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{{DISPLAYTITLE:L<sup>''p''</sup> space}} | |||
In [[mathematics]], the '''L<sup>''p''</sup> spaces''' are [[function space]]s defined using a natural generalization of the ''p''-norm for finite-dimensional [[vector space]]s. They are sometimes called '''Lebesgue spaces''', named after [[Henri Lebesgue]] {{harv|Dunford|Schwartz|1958|loc=III.3}}, although according to the [[Nicolas Bourbaki|Bourbaki]] group {{harv|Bourbaki|1987}} they were first introduced by [[Frigyes Riesz]] {{harv|Riesz|1910}}. | |||
'''L<sup>''p''</sup> spaces''' form an important class of [[Banach space]]s in [[functional analysis]], and of [[topological vector space]]s. | |||
Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines. | |||
==The ''p''-norm in finite dimensions== | |||
[[Image:Vector norms.svg|frame|right|Illustrations of [[unit circle]]s in different ''p''-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding ''p'').]] | |||
[[Image:Superellipse rounded diamond.svg|thumb|left|Unit circle ([[superellipse]]) in ''p'' = {{frac|3|2}} norm]] | |||
= | The length of a vector ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>) in the ''n''-dimensional [[real number|real]] [[vector space]] '''R'''<sup>''n''</sup> is usually given by the [[Euclidean norm]]: | ||
:<math>\ \|x\|_2=\left(x_1^2+x_2^2+\dotsb+x_n^2\right)^{\frac{1}{2}}</math> | |||
The Euclidean distance between two points ''x'' and ''y'' is the length <math>\scriptstyle \|x \,-\, y\|_2</math> of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. For example, taxi drivers in Manhattan should measure distance not in terms of the length of the straight line to their destination, but in terms of the [[Manhattan distance]], which takes into account that streets are either orthogonal or parallel to each other. The class of ''p''-norms generalizes these two examples and has an abundance of applications in many parts of [[mathematics]], [[physics]], and [[computer science]]. | |||
== | === Definition === | ||
For a [[real number]] ''p'' ≥ 1, the '''''p''-norm''' or '''''L''<sup>''p''</sup>-norm''' of ''x'' is defined by | |||
:<math>\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}}</math> | |||
The Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the [[Manhattan distance]]. | |||
= | The '''''L''<sup>∞</sup>-norm''' or [[Chebyshev distance|maximum norm]] (or uniform norm) is the limit of the ''L''<sup>''p''</sup>-norms for <math>\scriptstyle p \,\to\, \infty</math>. It turns out that this limit is equivalent to the following definition: | ||
:<math>\ \|x\|_\infty=\max \left\{|x_1|, |x_2|, \dotsc, |x_n|\right\}</math> | |||
For all ''p'' ≥ 1, the p-norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or [[norm (mathematics)|norm]]), which are that: | |||
* only the zero vector has zero length, | |||
* the length of the vector is positive homogeneous with respect to multiplication by a scalar, and | |||
* the length of the sum of two vectors is no larger than the sum of lengths of the vectors ([[triangle inequality]]). | |||
Abstractly speaking, this means that '''R'''<sup>''n''</sup> together with the ''p''-norm is a [[Banach space]]. This Banach space is the '''''L''<sup>''p''</sup>-space''' over '''R'''<sup>''n''</sup>. | |||
==== Relations between ''p''-norms ==== | |||
It is intuitively clear that the grid distance ("Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). | |||
Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm: | |||
:<math>\|x\|_2 \leq \|x\|_1</math> | |||
This fact generalizes to ''p''-norms in that the ''p''-norm <math>\scriptstyle \|x\|_p</math> of any given vector ''x'' does not grow with ''p'': | |||
:<math>\|x\|_{p+a} \leq \|x\|_{p}</math> for any vector ''x'' and real numbers ''p'' ≥ 1 and ''a'' ≥ 0. (In fact this remains true for 1>''p''>0 and ''a'' ≥ 0.) | |||
For the opposite direction, the following relation between the 1-norm and the 2-norm is known: | |||
:<math>\|x\|_1 \leq \sqrt{n}\|x\|_2</math> | |||
This inequality depends on the dimension ''n'' of the underlying vector space and follows directly from the [[Cauchy–Schwarz inequality]]. | |||
In general, for vectors in <math>\mathbb{C}^n</math> where p > r > 0: | |||
:<math>\|x\|_p\leq\|x\|_r\leq n^{\left(\frac{1}{r} - \frac{1}{p}\right)}\|x\|_p</math> | |||
=== When 0 < ''p'' < 1 === | |||
[[Image:Astroid.svg|thumb|right|[[Astroid]], unit circle in ''p'' = {{frac|2|3}} metric]] | |||
In '''R'''<sup>''n''</sup> for ''n'' > 1, the formula | |||
:<math>\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}}</math> | |||
defines an absolutely [[homogeneous function]] of degree 1 for 0 < ''p'' < 1; however, the resulting function does not define an [[F-norm]], because it is not [[Subadditivity|subadditive]]. In '''R'''<sup>''n''</sup> for ''n'' > 1, the formula for 0 < ''p'' < 1 | |||
:<math> |x_1|^p + |x_2|^p + \dotsb + |x_n|^p</math> | |||
defines a subadditive function, which does define an F-norm. This F-norm is homogeneous of degree p. | |||
However, the function | |||
:<math>d_p(x,y) = \sum_{i=1}^n |x_i-y_i|^p</math> | |||
defines a [[metric space|metric]]. The metric space ('''R'''<sup>''n''</sup>, ''d''<sub>''p''</sub>) is denoted by ℓ<sub>''n''</sub><sup>''p''</sup>. | |||
Although the ''p''-unit ball ''B''<sub>''n''</sub><sup>''p''</sup> around the origin in this metric is "concave", the topology defined on '''R'''<sup>''n''</sup> by the metric ''d''<sub>''p''</sup> is the usual vector space topology of '''R'''<sup>''n''</sup>, hence ℓ<sub>''n''</sub><sup>''p''</sup> is a [[locally convex]] topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓ<sub>''n''</sub><sup>''p''</sup> is to denote by ''C''<sub>''p''</sub>(''n'') the smallest constant ''C'' such that the multiple ''C'' ''B''<sub>''n''</sub><sup>''p''</sup> of the ''p''-unit ball contains the convex hull of ''B''<sub>''n''</sub><sup>''p''</sup>, equal to ''B''<sub>''n''</sub><sup>1</sup>. The fact that ''C''<sub>''p''</sub>(''n'') = ''n''<sup>1/''p'' – 1</sup> tends to infinity with ''n'' (for fixed ''p'' < 1) reflects the fact that the infinite-dimensional sequence space ℓ<sup>''p''</sup> defined below, is no longer locally convex. | |||
===When ''p'' = 0=== | |||
There is one l<sub>0</sub> norm and another function called the l<sub>0</sub> "norm" (with quotation marks). | |||
The mathematical definition of the l<sub>0</sub> norm was established by [[Banach]]'s ''[[Theory of Linear Operations]]''. The [[F-space|space]] of sequences has a complete metric topology provided by the [[F-space|F-norm]] <math>\scriptstyle (x_n) \,\mapsto\, \sum_n{2^{-n} |x_n|/(1 \,+\, |x_n| )}</math>, which is discussed by Stefan Rolewicz in ''Metric Linear Spaces''.<ref name="RolewiczControl">{{Citation | title=Functional analysis and control theory: Linear systems|last=Rolewicz |first=Stefan|year=1987| isbn=90-277-2186-6| publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers|oclc=13064804|edition=Translated from the Polish by Ewa Bednarczuk|series=Mathematics and its Applications (East European Series)|location=Dordrecht; Warsaw|volume=29|pages=xvi+524| mr=920371}}</ref> The l<sub>0</sub>-normed space is studied in functional analysis, probability theory, and harmonic analysis. | |||
Another function was called the l<sub>0</sub> "norm" by [[David Donoho]] — whose quotation marks warn that this function is not a proper norm — is the number of non-zero entries of the vector ''x''. Many authors [[abuse of terminology|abuse terminology]] by omitting the quotation marks. Defining 0<sup>0</sup> = 0, the zero "norm" of ''x'' is equal to <math>\scriptstyle |x_1|^0 \,+\, |x_2|^0 \,+\, \dotsb \,+\, |x_n|^0</math>. This is not a [[norm (mathematics)|norm]] (B-norm, with "B" for [[Banach]]) because it is not homogeneous. Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in [[scientific computing]], [[information theory]], and [[statistics]] – notably in [[compressed sensing]] in [[signal processing]] and computational [[harmonic analysis]]. | |||
==The ''p''-norm in countably infinite dimensions== | |||
:{{Details|Sequence space}} | |||
The ''p''-norm can be extended to vectors that have an infinite number of components, which yields the space <math>\scriptstyle \ell^p</math>. This contains as special cases: | |||
* <math>\scriptstyle \ell^1</math>, the space of sequences whose series is [[Absolute convergence|absolutely convergent]], | |||
* <math>\scriptstyle \ell^2</math>, the space of '''square-summable''' sequences, which is a [[Hilbert space]], and | |||
* <math>\scriptstyle \ell^\infty</math>, the space of [[bounded sequence]]s. | |||
The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. | |||
Explicitly, for <math>\scriptstyle \ x \;=\; (x_1,\, x_2,\, \dotsc,\, x_n,\, x_{n+1},\, \dotsc)</math> an infinite [[sequence]] of real (or [[complex number|complex]]) numbers, define the vector sum to be | |||
:<math>\begin{align} | |||
&(x_1, x_2, \dotsc, x_n, x_{n+1},\dotsc)+(y_1, y_2, \dotsc, y_n, y_{n+1},\dotsc) =\\ | |||
&(x_1+y_1, x_2+y_2, \dotsc, x_n+y_n, x_{n+1}+y_{n+1},\dotsc) | |||
\end{align}</math> | |||
while the scalar action is given by | |||
:<math>\lambda(x_1, x_2, \dotsc, x_n, x_{n+1},\dotsc) = (\lambda x_1, \lambda x_2, \dotsc, \lambda x_n, \lambda x_{n+1},\dotsc)</math> | |||
Define the ''p''-norm | |||
:<math>\|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb+|x_n|^p + |x_{n+1}|^p + \dotsb\right)^{\frac{1}{p}}</math> | |||
Here, a complication arises, namely that the [[series (mathematics)|series]] on the right is not always convergent, so for example, the sequence made up of only ones, (1, 1, 1, …), will have an infinite ''p''-norm (length) for every finite ''p'' ≥ 1. The space ℓ<sup>''p''</sup> is then defined as the set of all infinite sequences of real (or complex) numbers such that the ''p''-norm is finite. | |||
One can check that as ''p'' increases, the set ℓ<sup>''p''</sup> grows larger. For example, the sequence | |||
:<math>\left(1, \frac{1}{2}, \dotsc, \frac{1}{n}, \frac{1}{n+1},\dotsc\right)</math> | |||
is not in ℓ<sup>1</sup>, but it is in ℓ<sup>''p''</sup> for ''p'' > 1, as the series | |||
:<math>1^p + \frac{1}{2^p} + \dotsb + \frac{1}{n^p} + \frac{1}{(n+1)^p}+\dotsb</math> | |||
diverges for ''p'' = 1 (the [[harmonic series (mathematics)|harmonic series]]), but is convergent for ''p'' > 1. | |||
One also defines the ∞-norm using the [[supremum]]: | |||
:<math>\ \|x\|_\infty=\sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \dotsc)</math> | |||
and the corresponding space ℓ<sup>∞</sup> of all bounded sequences. It turns out that<ref>{{Citation | last1=Maddox | first1=I.J. | author1-link=I.J. Maddox | title=Elements of Functional Analysis | publisher=CUP | location=Cambridge | edition=2nd | year=1988}}, page 16</ref> | |||
:<math>\ \|x\|_\infty = \lim_{p\to\infty}\|x\|_p</math> | |||
if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider ℓ<sup>''p''</sup> spaces for 1 ≤ ''p'' ≤ ∞. | |||
The ''p''-norm thus defined on ℓ<sup>''p''</sup> is indeed a norm, and ℓ<sup>''p''</sup> together with this norm is a [[Banach space]]. The fully general ''L''<sup>''p''</sup> space is obtained — as seen below — by considering vectors, not only with finitely or countably-infinitely many components, but with "''arbitrarily many components''"; in other words, [[function (mathematics)|functions]]. An [[integral]] instead of a sum is used to define the ''p''-norm. | |||
== ''L<sup>p</sup>'' spaces == | |||
Let 1 ≤ ''p'' < ∞ and (''S'', ''Σ'', ''μ'') be a [[measure space]]. Consider the set of all [[measurable function]]s from ''S'' to '''C''' (or '''R''') whose [[absolute value]] raised to the ''p''-th power has finite integral, or equivalently, that | |||
:<math>\|f\|_p \equiv \left({\int_S |f|^p\;\mathrm{d}\mu}\right)^{\frac{1}{p}}<\infty</math> | |||
The set of such functions forms a vector space, with the following natural operations: | |||
:<math>(f+g)(x) = f(x)+g(x), \ \ \ \text{and} \ \ \ (\lambda f)(x) = \lambda f(x) \,</math> | |||
for every scalar ''λ''. | |||
That the sum of two ''p''<sup>th</sup> power integrable functions is again ''p''<sup>th</sup> power integrable follows from the inequality |''f'' + ''g''|<sup>''p''</sup> ≤ 2<sup>''p-1''</sup> (|''f''|<sup>''p''</sup> + |''g''|<sup>''p''</sup>). In fact, more is true. [[Minkowski inequality|Minkowski's inequality]] says the [[triangle inequality]] holds for || · ||<sub>''p''</sub>. Thus the set of ''p''<sup>th</sup> power integrable functions, together with the function || · ||<sub>''p''</sub>, is a [[seminorm]]ed vector space, which is denoted by <math>\scriptstyle \mathcal{L}^p(S,\, \mu)</math>. | |||
{{anchor|kernel}} | |||
This can be made into a normed vector space in a standard way; one simply takes the [[Quotient Space|quotient space]] with respect to the [[kernel (set theory)|kernel]] of || · ||<sub>''p''</sub>. Since for any measurable function ''f'', we have that ||''f''||<sub>''p''</sub> = 0 if and only if ''f'' = 0 [[almost everywhere]], the kernel of || · ||<sub>''p''</sub> does not depend upon ''p'', | |||
:<math>N \equiv \mathrm{ker}(\|\cdot\|_p) = \{f : f = 0 \ \mu\text{-almost everywhere} \}</math> | |||
In the quotient space, two functions ''f'' and ''g'' are identified if ''f'' = ''g'' almost everywhere. The resulting normed vector space is, by definition, | |||
:<math>L^p(S, \mu) \equiv \mathcal{L}^p(S, \mu) / N</math> | |||
For ''p'' = ∞, the space ''L''<sup>∞</sup>(''S'', ''μ'') is defined as follows. We start with the set of all measurable functions from ''S'' to '''C''' (or '''R''') which are '''essentially bounded''', i.e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by ''L''<sup>∞</sup>(''S'', ''μ''). For ''f'' in ''L''<sup>∞</sup>(''S'', ''μ''), its [[essential supremum]] serves as an appropriate norm: | |||
:<math>\|f\|_\infty \equiv \inf \{ C\ge 0 : |f(x)| \le C \mbox{ for almost every } x\}.</math> | |||
As before, we have | |||
:<math>\|f\|_\infty=\lim_{p\to\infty}\|f\|_p</math> | |||
if ''f'' ∈ ''L''<sup>∞</sup>(''S'', ''μ'') ∩ ''L''<sup>''q''</sup>(''S'', ''μ'') for some ''q'' < ∞. | |||
For 1 ≤ ''p'' ≤ ∞, ''L''<sup>''p''</sup>(''S'', ''μ'') is a [[Banach space]]. The fact that ''L''<sup>''p''</sup> is ''complete'' is often referred to as the ''[[Riesz-Fischer theorem]]''. Completeness can be checked using the convergence theorems for Lebesgue integrals. | |||
When the underlying measure space ''S'' is understood, ''L''<sup>''p''</sup>(''S'', ''μ'') is often abbreviated ''L''<sup>''p''</sup>(''μ''), or just ''L''<sup>''p''</sup>. The above definitions generalize to [[Bochner space]]s. | |||
=== Special cases === | |||
When ''p'' = 2; like the ℓ<sup>2</sup> space, the space ''L''<sup>2</sup> is the only [[Hilbert space]] of this class. In the complex case, the inner product on ''L''<sup>2</sup> is defined by | |||
:<math> \langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x)</math> | |||
The additional inner product structure allows for a richer theory, with applications to, for instance, [[Fourier series]] and [[quantum mechanics]]. Functions in ''L''<sup>2</sup> are sometimes called '''[[quadratically integrable function]]s''', '''square-integrable functions''' or '''square-summable functions''', but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a [[Riemann integral]] {{harv|Titchmarsh|1976}}. | |||
If we use complex-valued functions, the space ''L''<sup>∞</sup> is a [[commutative]] [[C*-algebra]] with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative [[von Neumann algebra]]. An element of ''L''<sup>∞</sup> defines a [[bounded operator]] on any ''L''<sup>''p''</sup> space by [[multiplication operator|multiplication]]. | |||
The ℓ<sup>''p''</sup> spaces (1 ≤ ''p'' ≤ ∞) are a special case of ''L<sup>p</sup>'' spaces, when ''S'' is the set '''N''' of positive [[integer]]s, and the measure ''μ'' is the [[counting measure]] on '''N'''. More generally, if one considers any set ''S'' with the counting measure, the resulting ''L<sup> p</sup>'' space is denoted ℓ<sup>''p''</sup>(''S''). For example, the space ℓ<sup>''p''</sup>('''Z''') is the space of all sequences indexed by the integers, and when defining the ''p''-norm on such a space, one sums over all the integers. The space ℓ<sup>''p''</sup>(''n''), where ''n'' is the set with ''n'' elements, is '''R'''<sup>''n''</sup> with its ''p''-norm as defined above. As any Hilbert space, every space ''L''<sup>2</sup> is linearly isometric to a suitable ℓ<sup>2</sup>(''I''), where the cardinality of the set ''I'' is the cardinality of an arbitrary Hilbertian basis for this particular ''L''<sup>2</sup>. | |||
==Properties of ''L''<sup>''p''</sup> spaces== | |||
===Dual spaces=== | |||
The [[Dual_space#Continuous_dual_space|dual space]] (the space of all continuous linear functionals) of ''L''<sup>''p''</sup>(''μ'') for 1 < ''p'' < ∞ has a natural isomorphism with ''L''<sup>''q''</sup>(''μ''), where ''q''  is such that 1/''p'' + 1/''q'' = 1, which associates ''g'' ∈ ''L''<sup>''q''</sup>(''μ'') with the functional ''κ''<sub>''p''</sub>(''g'') ∈ ''L''<sup>''p''</sup>(''μ'')<sup>∗</sup> defined by | |||
:<math>\kappa_p(g) \colon f \in L^p(\mu) \mapsto \int f g \, \mathrm{d}\mu</math> | |||
The fact that ''κ''<sub>''p''</sub>(''g'') is well defined and continuous follows from [[Hölder's inequality]]. The mapping ''κ''<sub>''p''</sub> is a linear mapping from ''L''<sup>''q''</sup>(''μ'') into ''L''<sup>''p''</sup>(''μ'')<sup>∗</sup>, which is an [[isometry]] by the [[Hölder's inequality#Extremal equality|extremal case]] of Hölder's inequality. It is also possible to show (for example with the [[Radon–Nikodym theorem]], see<ref>{{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and Complex Analysis | publisher=Tata McGraw-Hill | location=New Delhi | edition=2nd | year=1980 |isbn=9780070542341}}, Theorem 6.16</ref>) that any ''G'' ∈ ''L''<sup>''p''</sup>(''μ'')<sup>∗</sup> can be expressed this way: i.e., that ''κ''<sub>''p''</sub> is ''onto''. Since ''κ''<sub>''p''</sub> is onto and isometric, it is an [[isomorphism]] of [[Banach space]]s. With this (isometric) isomorphism in mind, it is usual to say simply that ''L''<sup>''q''</sup> "''is''" the dual of ''L''<sup>''p''</sup>. | |||
When 1 < ''p'' < ∞, the space ''L''<sup>''p''</sup>(''μ'') is [[reflexive space|reflexive]]. Let ''κ''<sub>''p''</sub> be the above map and let ''κ''<sub>''q''</sub> be the corresponding linear isometry from ''L''<sup>''p''</sup>(''μ'') onto ''L''<sup>''q''</sup>(''μ'')<sup>∗</sup>. The map | |||
:<math>j_p \colon L^p(\mu) \overset{\kappa_q}{\to} L^q(\mu)^* \overset{\,\,\left(\kappa_p^{-1}\right)^*}{\longrightarrow} L^p(\mu)^{**}</math> | |||
from ''L''<sup>''p''</sup>(''μ'') to ''L''<sup>''p''</sup>(''μ'')<sup>∗∗</sup>, obtained by composing ''κ''<sub>''q''</sub> with the [[Dual space#Transpose of a continuous linear map|transpose]] (or adjoint) of the inverse of ''κ''<sub>''p''</sub>, coincides with the [[Reflexive space#Definitions|canonical embedding]] ''J''  of ''L''<sup>''p''</sup>(''μ'') into its bidual. Moreover, the map ''j''<sub>''p''</sub> is onto, as composition of two onto isometries, and this proves reflexivity. | |||
If the measure ''μ'' on ''S'' is [[sigma-finite]], then the dual of ''L''<sup>1</sup>(''μ'') is isometrically isomorphic to ''L''<sup>∞</sup>(''μ'') (more precisely, the map ''κ''<sub>1</sub> corresponding to ''p'' = 1 is an isometry from ''L''<sup>∞</sup>(''μ'') onto ''L''<sup>1</sup>(''μ'')<sup>∗</sup>). | |||
The dual of ''L''<sup>∞</sup> is subtler. Elements of (''L''<sup>∞</sup>(''μ''))<sup>∗</sup> can be identified with bounded signed ''finitely'' additive measures on ''S'' that are [[absolutely continuous]] with respect to ''μ''. See [[ba space]] for more details. If we assume the axiom of choice, this space is much bigger than ''L''<sup>1</sup>(''μ'') except in some trivial cases. However, [[Saharon Shelah]] proved that there are relatively consistent extensions of [[Zermelo-Fraenkel set theory]] (ZF + [[Axiom of dependent choice|DC]] + "Every subset of the real numbers has the [[Baire property]]") in which the dual of ''ℓ''<sup>∞</sup> is ''ℓ''<sup>1</sup>. <ref>{{Citation | title=Handbook of Analysis and its Foundations|last=Schechter |first=Eric|year=1997| publisher=Academic Press Inc.|location=London}} See Sections 14.77 and 27.44--47</ref> | |||
===Embeddings=== | |||
Colloquially, if 1 ≤ ''p'' < ''q'' ≤ ∞, ''L<sup>p</sup>''(''S'', ''μ'') contains functions that are more locally singular, while elements of ''L<sup>q</sup>''(''S'', ''μ'') can be more spread out. Consider the Lebesgue measure on the half line (0, ∞). A continuous function in ''L''<sup>1</sup> might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in ''L''<sup>∞</sup> need not decay at all but no blow-up is allowed. The precise technical result is the following: | |||
#Let 0 ≤ ''p'' < ''q'' ≤ ∞. ''L<sup>q</sup>''(''S'', ''μ'') is contained in ''L<sup>p</sup>''(''S'', μ) iff ''S'' does not contain sets of arbitrarily large measure, and | |||
#Let 0 ≤ ''p'' < ''q'' ≤ ∞. ''L<sup>p</sup>''(''S'', ''μ'') is contained in ''L<sup>q</sup>''(''S'', ''μ'') iff ''S'' does not contain sets of arbitrarily small non-zero measure. | |||
In particular, if the domain ''S'' has finite measure, the bound (a consequence of [[Jensen's inequality]]) | |||
:<math>\ \|f\|_p \le \mu(S)^{\frac{1}{p} - \frac{1}{q}} \|f\|_q </math> | |||
means the space ''L''<sup>''q''</sup> is continuously embedded in ''L''<sup>''p''</sup>. That is to say, the identity operator is a bounded linear map from ''L''<sup>''q''</sup> to ''L''<sup>''p''</sup>. The constant appearing in the above inequality is optimal, in the sense that the [[operator norm]] of the identity ''I'' : ''L<sup>q</sup>''(''S'', ''μ'') → ''L<sup>p</sup>''(''S'', ''μ'') is precisely | |||
:<math>\|I\|_{q,p} = \mu(S)^{\frac{1}{p} - \frac{1}{q}}</math> | |||
the case of equality being achieved exactly when ''f'' = 1 a.e.[μ]. | |||
=== Dense subspaces === | |||
It is assumed that 1 ≤ ''p'' < ∞ throughout this section.<br /> | |||
Let (''S'', ''Σ'', ''μ'') be a measure space. An ''integrable simple function'' ''f''  on ''S''  is one of the form | |||
:<math>f = \sum_{j=1}^n a_j \mathbf{1}_{A_j}</math> | |||
where ''a<sub>j</sub>'' is scalar and ''A<sub>j</sub>'' ∈ ''Σ''  has finite measure, for ''j'' = 1, …, ''n''. By construction of the [[Lebesgue integration|integral]], the vector space of integrable simple functions is dense in ''L''<sup>''p''</sup>(''S'', ''Σ'', ''μ''). | |||
More can be said when ''S''  is a [[Metrization theorem|metrizable]] [[topological space]] and ''Σ''  its [[Borel algebra|Borel ''σ''–algebra]], ''i.e.'', the smallest ''σ''–algebra of subsets of ''S''  containing the [[open set]]s. | |||
Suppose that ''V'' ⊂ ''S''  is an open set with ''μ''(''V'') < ∞. It can be proved that for every Borel set ''A'' ∈ ''Σ''  contained in ''V'', and for every ''ε'' > 0, there exist a closed set ''F''  and an open set ''U''  such that | |||
:<math>F \subset A \subset U \subset V \ \ \text{and} \ \ \mu(U) - \mu(F) = \mu(U \setminus F) < \varepsilon</math> | |||
It follows that there exists ''φ'' continuous on ''S''  such that | |||
:<math>0 \le \varphi \le \mathbf{1}_V \ \text{and} \ \int_S |\mathbf{1}_A - \varphi| \, \mathrm{d}\mu < \varepsilon</math> | |||
If ''S''  can be covered by an increasing sequence (''V<sub>n</sub>'') of open sets that have finite measure, then the space of ''p''–integrable continuous functions is dense in ''L''<sup>''p''</sup>(''S'', ''Σ'', ''μ''). More precisely, one can use bounded continuous functions that vanish outside one of the open sets ''V<sub>n</sub>''. | |||
This applies in particular when ''S'' = '''R'''<sup>''d''</sup> and when ''μ'' is the Lebesgue measure. The space of continuous and compactly supported functions is dense in ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>). Similarly, the space of integrable ''step functions''  is dense in ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>); this space is the linear span of indicator functions of bounded intervals when ''d'' = 1, of bounded rectangles when ''d'' = 2 and more generally of products of bounded intervals.<br /> | |||
Several properties of general functions in ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>) are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>), in the following sense: for every ''f'' ∈ ''L''<sup>''p''</sup>('''R'''<sup>''d''</sup>), | |||
:<math>\|\tau_t f - f \|_p \rightarrow 0</math> | |||
when ''t'' ∈ '''R'''<sup>''d''</sup> tends to 0, where <math>\scriptstyle \tau_t f</math> is the translated function defined by <math>\scriptstyle (\tau_t f)(x) \;=\; f(x \,-\, t)</math>. | |||
==Applications== | |||
''L<sup>p</sup>'' spaces are widely used in mathematics and applications. | |||
===Hausdorff–Young inequality=== | |||
The [[Fourier transform]] for the real line (resp. for periodic functions, cf. [[Fourier series]]) maps ''L<sup>p</sup>''('''R''') to ''L<sup>q</sup>''('''R''') (resp. ''L<sup>p</sup>''('''T''') to ℓ<sup>''q''</sup>), where 1 ≤ ''p'' ≤ 2 and 1/''p'' + 1/''q'' = 1. This is a consequence of the [[Riesz-Thorin theorem|Riesz-Thorin interpolation theorem]], and is made precise with the [[Hausdorff–Young inequality]]. | |||
By contrast, if ''p'' > 2, the Fourier transform does not map into ''L<sup>q</sup>''. | |||
===Hilbert spaces=== | |||
[[Hilbert space]]s are central to many applications, from [[quantum mechanics]] to [[stochastic calculus]]. The spaces ''L''<sup>2</sup> and ℓ<sup>2</sup> are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ<sup>2</sup>(''E''), where ''E'' is a set with an appropriate cardinality. | |||
===Statistics=== | |||
In [[statistics]], measures of [[central tendency]] and [[statistical dispersion]], such as the [[mean]], [[median]], and [[standard deviation]], are defined in terms of ''L''<sup>''p''</sup> metrics, and measures of central tendency can be characterized as [[Average#Solutions to variational problems|solutions to variational problems]]. | |||
== ''L''<sup>''p''</sup> for 0 < ''p'' < 1 == | |||
Let (''S'', ''Σ'', ''μ'') be a measure space. If 0 < ''p'' < 1, then ''L<sup>p</sup>''(''μ'') can be defined as above: it is the vector space of those measurable functions ''f'' such that | |||
:<math>N_p(f) = \int_S |f|^p\, d\mu < \infty</math>. | |||
As before, we may introduce the ''p''-norm || ''f'' ||<sub>''p''</sub> = ''N''<sub>''p''</sup>(''f'')<sup>1/''p''</sup>, | |||
but || · ||<sub>''p''</sub> does not satisfy the triangle inequality in this case, and defines only a [[quasi-norm]]. | |||
The inequality (''a'' + ''b'')<sup>''p''</sup> ≤ ''a''<sup>''p''</sup> + ''b''<sup>''p''</sup>, valid for ''a'' ≥ 0 and ''b'' ≥ 0 implies that {{harv|Rudin|1991|loc=§1.47}} | |||
:<math>N_p(f+g)\le N_p(f) + N_p(g)</math> | |||
and so the function | |||
:<math>d_p(f,g) = N_p(f-g) = \|f - g\|_p^p</math> | |||
is a metric on ''L''<sup>''p''</sup>(''μ''). The resulting metric space is [[complete space|complete]]; the verification is similar to the familiar case when ''p'' ≥ 1. | |||
In this setting ''L''<sup>''p''</sup> satisfies a ''reverse Minkowski inequality'', that is for ''u'' and ''v'' in ''L<sup>p</sup>'' | |||
:<math>\|\,|u|+|v|\,\|_p\geq \|u\|_p+\|v\|_p</math> | |||
This result may be used to prove Clarkson's inequalities, which are in turn used to establish the [[Uniformly convex space|uniform convexity]] of the spaces ''L''<sup>''p''</sup> | |||
for 1 < ''p'' < ∞ {{harv|Adams|Fournier|2003}}. | |||
The space ''L''<sup>''p''</sup> for 0 < ''p'' < 1 is an [[F-space]]: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is also [[locally bounded]], much like the case ''p'' ≥ 1. It is the prototypical example of an [[F-space]] that, for most reasonable measure spaces, is not [[locally convex]]: in ℓ<sup>''p''</sup> or | |||
''L''<sup>''p''</sup>([0, 1]), every open convex set containing the 0 function is unbounded for the ''p''-quasi-norm; therefore, the 0 vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space ''S'' contains an infinite family of disjoint measurable sets of finite positive measure. | |||
The only nonempty convex open set in ''L''<sup>''p''</sup>([0, 1]) is the entire space {{harv|Rudin|1991|loc=§1.47}}. As a particular consequence, there are no nonzero linear functionals on ''L''<sup>''p''</sup>([0, 1]): the dual space is the zero space. In the case of the [[counting measure]] on the natural numbers (producing the sequence space ''L''<sup>''p''</sup>(''μ'') = ℓ<sup>''p''</sup>), the bounded linear functionals on ℓ<sup>''p''</sup> are exactly those that are bounded on ℓ<sup>1</sup>, namely those given by sequences in ℓ<sup>∞</sup>. Although ℓ<sup>''p''</sup> does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology. | |||
The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on '''R'''<sup>''n''</sup>, rather than work with ''L''<sup>''p''</sup> for 0 < ''p'' < 1, it is common to work with the [[Hardy space]] ''H''<sup>''p''</sup> whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the [[Hahn–Banach theorem]] still fails in ''H''<sup>''p''</sup> for ''p'' < 1 {{harv|Duren|1970|loc=§7.5}}. | |||
=== ''L''<sup>0</sup>, the space of measurable functions === | |||
The vector space of (equivalence classes of) measurable functions on (''S'', ''Σ'', ''μ'') is denoted ''L''<sup>0</sup>(''S'', ''Σ'', ''μ'') {{harv|Kalton|Peck|Roberts|1984}}. By definition, it contains all the ''L''<sup>''p''</sup>, and is equipped with the topology of [[Convergence in measure|''convergence in measure'']]. When ''μ'' is a probability measure (i.e., ''μ''(''S'') = 1), this mode of convergence is named [[Convergence in probability|''convergence in probability'']]. | |||
The description is easier when ''μ'' is finite. | |||
If ''μ'' is a finite measure on (''S'', ''Σ''), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods | |||
:<math>V_\varepsilon = \Bigl\{ f : \mu \bigl(\{x : |f(x)| > \varepsilon \} \bigr) < \varepsilon \Bigr\}, \ \ \varepsilon > 0</math> | |||
The topology can be defined by any metric ''d''  of the form | |||
:<math>d(f, g) = \int_S \varphi \bigl( |f(x) - g(x)| \bigr) \, \mathrm{d}\mu(x)</math> | |||
where ''φ''  is bounded continuous concave and non-decreasing on [0, ∞), with ''φ''(0) = 0 and ''φ''(''t'') > 0 when ''t'' > 0 (for example, ''φ''(''t'') = min(''t'', 1)). Such a metric is called ''[[Paul Lévy (mathematician)|Lévy]]-metric for L<sup>0</sup>.'' Under this metric the space ''L''<sup>0</sup> is complete (it is again an F-space). The space ''L''<sup>0</sup> is in general not locally bounded, and not locally convex. | |||
For the infinite Lebesgue measure ''λ'' on '''R'''<sup>''n''</sup>, the definition of the fundamental system of neighborhoods could be modified as follows | |||
:<math>W_\varepsilon = \left\{ f : \lambda \left(\left\{ x : |f(x)| > \varepsilon \ \text{and} \ |x| < \frac{1}{\varepsilon}\right\} \right) < \varepsilon \right\}</math> | |||
The resulting space ''L''<sup>0</sup>('''R'''<sup>''n''</sup>, ''λ'') coincides as topological vector space with ''L''<sup>0</sup>('''R'''<sup>''n''</sup>, ''g''(''x'') d''λ''(x)), for any positive ''λ''–integrable density ''g''. | |||
==Weak ''L<sup>p</sup>''== | |||
Let (''S'', ''Σ'', ''μ'') be a measure space, and ''f'' a [[measurable function]] with real or complex values on ''S''. The [[cumulative distribution function|distribution function]] of ''f'' is defined for ''t'' > 0 by | |||
:<math>\lambda_f(t) = \mu\left\{x\in S: |f(x)| > t\right\}</math> | |||
If ''f'' is in ''L''<sup>''p''</sup>(''S'', ''μ'') for some ''p'' with 1 ≤ ''p'' < ∞, then by [[Markov's inequality]], | |||
:<math>\lambda_f(t)\le \frac{\|f\|_p^p}{t^p}</math> | |||
A function ''f'' is said to be in the space '''weak ''L<sup>p</sup>''(''S'', ''μ'')''', or ''L<sup>p,w</sup>''(''S'', ''μ''), if there is a constant ''C'' > 0 such that, for all ''t'' > 0, | |||
:<math>\lambda_f(t) \le \frac{C^p}{t^p}</math> | |||
The best constant ''C'' for this inequality is the ''L<sup>p,w</sup>''-norm of ''f'', and is denoted by | |||
:<math>\|f\|_{p,w} = \sup_{t > 0} ~ t \lambda_f^{\frac{1}{p}}(t)</math> | |||
The weak ''L''<sup>''p''</sup> coincide with the [[Lorentz space]]s ''L''<sup>''p'',∞</sup>, so this notation is also used to denote them. | |||
The ''L<sup>p,w</sup>''-norm is not a true norm, since the [[triangle inequality]] fails to hold. Nevertheless, for ''f'' in ''L''<sup>p</sup>(''S'', ''μ''), | |||
:<math>\|f\|_{p,w}\le \|f\|_p</math> | |||
and in particular ''L<sup>p</sup>''(''S'', ''μ'') ⊂ ''L<sup>p,w</sup>''(''S'', ''μ''). Under the convention that two functions are equal if they are equal ''μ'' almost everywhere, then the spaces ''L''<sup>p,w</sup> are complete {{harv|Grafakos|2004}}. | |||
For any 0 < ''r'' < ''p'' the expression | |||
:<math>||| f |||_{L^{p,\infty}}=\sup_{0<\mu(E)<\infty} \mu(E)^{-\frac{1}{r}+\frac{1}{p}}\left(\int_E |f|^r\,d\mu\right)^{\frac{1}{r}}</math> | |||
is comparable to the ''L<sup>p,w</sup>''-norm. Further in the case ''p'' > 1, this expression defines a norm if ''r'' = 1. Hence for ''p'' > 1 the weak ''L''<sup>''p''</sup> spaces are [[Banach space]]s {{harv|Grafakos|2004}}. | |||
A major result that uses the ''L<sup>p,w</sup>''-spaces is the [[Marcinkiewicz interpolation|Marcinkiewicz interpolation theorem]], which has broad applications to [[harmonic analysis]] and the study of [[singular integrals]]. | |||
==Weighted ''L<sup>p</sup>'' spaces== | |||
As before, consider a [[measure space]] (''S'', ''Σ'', ''μ''). Let <math>\scriptstyle w :\; S \,\to\, [0,\, + \infty)</math> be a measurable function. The ''w''-'''weighted ''L<sup>p</sup>'' space''' is defined as ''L<sup>p</sup>''(''S'', ''w'' d''μ''), where ''w'' d''μ'' means the measure ''ν'' defined by | |||
:<math>\ \nu (A) \equiv \int_{A} w(x) \, \mathrm{d} \mu (x), \ \ \ A \in \Sigma</math> | |||
or, in terms of the [[Radon–Nikodym theorem|Radon–Nikodym derivative]], | |||
:<math>\ w = \frac{\mathrm{d} \nu}{\mathrm{d} \mu}</math> | |||
The [[norm (mathematics)|norm]] for ''L<sup>p</sup>''(''S'', ''w'' d''μ'') is explicitly | |||
:<math>\ \| u \|_{L^{p} (S, w \, \mathrm{d} \mu)} \equiv \left( \int_{S} w(x) | u(x) |^{p} \, \mathrm{d} \mu (x) \right)^{\frac{1}{p}}</math> | |||
As ''L''<sup>''p''</sub>-spaces, the weighted spaces have nothing special, since ''L<sup>p</sup>''(''S'', ''w'' d''μ'') is equal to ''L''<sup>''p''</sup>(''S'', d''ν''). But they are the natural framework for several results in harmonic analysis {{harv|Grafakos|2004}}<!--Please check this reference. Appears in Grafakos "Modern Fourier analysis", Chapter 9.-->; they appear for example in the [[Muckenhoupt weights|Muckenhoupt theorem]]: for 1 < ''p'' < ∞, the classical [[Hilbert transform]] is defined on ''L''<sup>''p''</sub>('''T''', ''λ'') where '''T''' denotes the unit circle and ''λ'' the Lebesgue measure; the (nonlinear) [[Hardy–Littlewood maximal operator]] is bounded on ''L''<sup>''p''</sub>('''R'''<sup>''n''</sup>, ''λ''). Muckenhoupt's theorem describes weights ''w'' such that the Hilbert transform remains bounded on ''L<sup>p</sup>''('''T''', ''w'' d''λ'') and the maximal operator on ''L<sup>p</sup>''('''R'''<sup>''n''</sup>, ''w'' d''λ''). | |||
==''L<sup>p</sup>'' spaces on manifolds== | |||
One may also define spaces <math>\scriptstyle L^p(M)</math> on a manifold, called the '''intrinsic ''L<sup>p</sup>'' spaces''' of the manifold, using [[Density on a manifold|densities]]. | |||
==See also== | |||
* [[Birnbaum–Orlicz space]] | |||
* [[Hardy space]] | |||
* [[Riesz–Thorin theorem]] | |||
* [[Hölder mean]] | |||
* [[Hölder space]] | |||
* [[Root mean square]] | |||
* [[Locally integrable function]] <math>\left(\scriptstyle L^1_{\text{loc}}\right)</math> | |||
* [[Pontryagin duality#Haar measure|<math>\scriptstyle L^p(G)</math> spaces over a locally compact group <math>G</math>]] | |||
* [[Minkowski distance]] | |||
==Notes== | |||
{{Reflist}} | |||
==References== | |||
* {{citation|last1=Adams|first1=Robert A.|last2=Fournier|first2=John F.|title=Sobolev Spaces|edition=Second|publisher=Academic Press|year=2003|isbn=978-0-12-044143-3}}. | |||
* {{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Topological vector spaces|series=Elements of mathematics|publisher= Springer-Verlag|publication-place=Berlin|year=1987|isbn=978-3-540-13627-9}}. | |||
* {{citation | last=DiBenedetto|first=Emmanuele|title=Real analysis|publisher=Birkhäuser|year=2002|isbn=3-7643-4231-5}}. | |||
* {{citation|last1=Dunford|first1=Nelson|last2=Schwartz|first2=Jacob T.|title=Linear operators, volume I|publisher=Wiley-Interscience|year=1958}}. | |||
*{{citation | |||
|last= Duren|first=P.|title=Theory of H<sup>p</sup>-Spaces|year=1970|publisher= Academic Press|publication-place= New York}} | |||
* {{citation|title=Classical and Modern Fourier Analysis | last=Grafakos | first=Loukas | publisher=Pearson Education, Inc. | pages=253–257 | year=2004 | isbn=0-13-035399-X}}. | |||
* {{citation|last1=Hewitt|first1=Edwin|last2=Stromberg|first2=Karl|title=Real and abstract analysis|publisher=Springer-Verlag|year=1965}}. | |||
* {{citation | |||
|last1=Kalton|first1=Nigel J.|author-link=Nigel Kalton | |||
|last2=Peck|first2=N. Tenney | |||
|last3=Roberts|first3=James W. | |||
| title = An F-space sampler | |||
| series = London Mathematical Society Lecture Note Series|volume=89 | |||
| publisher = Cambridge University Press| publication-place = Cambridge | |||
| year = 1984 | isbn = 0-521-27585-7|mr=808777}} | |||
* {{citation | |||
|last=Riesz|first=Frigyes|authorlink=Frigyes Riesz | |||
|title=Untersuchungen über Systeme integrierbarer Funktionen|journal=Mathematische Annalen|volume=69|year=1910|pages=449–497 | |||
|doi=10.1007/BF01457637 | |||
|issue=4}} | |||
* {{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Functional Analysis | publisher=McGraw-Hill Science/Engineering/Math | isbn=978-0-07-054236-5 | year=1991}} | |||
* {{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and complex analysis | publisher=[[McGraw-Hill]] | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 | year=1987}} | |||
* {{citation|first=EC|last=Titchmarsh|authorlink=Edward Charles Titchmarsh|title=The theory of functions|publisher=Oxford University Press|year=1976|isbn=978-0-19-853349-8}} | |||
==External links== | |||
* {{springer|title=Lebesgue space|id=p/l057910}} | |||
* {{planetmath reference|id=6270|title=Proof that ''L''<sup>''p''</sup> spaces are complete }} | |||
{{DEFAULTSORT:Lp Space}} | |||
[[Category:Normed spaces]] | |||
[[Category:Banach spaces]] | |||
[[Category:Mathematical series]] | |||
[[Category:Function spaces]] |
Revision as of 17:31, 21 January 2014
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue Template:Harv, although according to the Bourbaki group Template:Harv they were first introduced by Frigyes Riesz Template:Harv. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.
The p-norm in finite dimensions
The length of a vector x = (x1, x2, …, xn) in the n-dimensional real vector space Rn is usually given by the Euclidean norm:
The Euclidean distance between two points x and y is the length of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. For example, taxi drivers in Manhattan should measure distance not in terms of the length of the straight line to their destination, but in terms of the Manhattan distance, which takes into account that streets are either orthogonal or parallel to each other. The class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.
Definition
For a real number p ≥ 1, the p-norm or Lp-norm of x is defined by
The Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the Manhattan distance.
The L∞-norm or maximum norm (or uniform norm) is the limit of the Lp-norms for . It turns out that this limit is equivalent to the following definition:
For all p ≥ 1, the p-norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that:
- only the zero vector has zero length,
- the length of the vector is positive homogeneous with respect to multiplication by a scalar, and
- the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).
Abstractly speaking, this means that Rn together with the p-norm is a Banach space. This Banach space is the Lp-space over Rn.
Relations between p-norms
It is intuitively clear that the grid distance ("Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:
This fact generalizes to p-norms in that the p-norm of any given vector x does not grow with p:
For the opposite direction, the following relation between the 1-norm and the 2-norm is known:
This inequality depends on the dimension n of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.
In general, for vectors in where p > r > 0:
When 0 < p < 1
In Rn for n > 1, the formula
defines an absolutely homogeneous function of degree 1 for 0 < p < 1; however, the resulting function does not define an F-norm, because it is not subadditive. In Rn for n > 1, the formula for 0 < p < 1
defines a subadditive function, which does define an F-norm. This F-norm is homogeneous of degree p.
However, the function
defines a metric. The metric space (Rn, dp) is denoted by ℓnp.
Although the p-unit ball Bnp around the origin in this metric is "concave", the topology defined on Rn by the metric dp is the usual vector space topology of Rn, hence ℓnp is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓnp is to denote by Cp(n) the smallest constant C such that the multiple C Bnp of the p-unit ball contains the convex hull of Bnp, equal to Bn1. The fact that Cp(n) = n1/p – 1 tends to infinity with n (for fixed p < 1) reflects the fact that the infinite-dimensional sequence space ℓp defined below, is no longer locally convex.
When p = 0
There is one l0 norm and another function called the l0 "norm" (with quotation marks).
The mathematical definition of the l0 norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm , which is discussed by Stefan Rolewicz in Metric Linear Spaces.[1] The l0-normed space is studied in functional analysis, probability theory, and harmonic analysis.
Another function was called the l0 "norm" by David Donoho — whose quotation marks warn that this function is not a proper norm — is the number of non-zero entries of the vector x. Many authors abuse terminology by omitting the quotation marks. Defining 00 = 0, the zero "norm" of x is equal to . This is not a norm (B-norm, with "B" for Banach) because it is not homogeneous. Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics – notably in compressed sensing in signal processing and computational harmonic analysis.
The p-norm in countably infinite dimensions
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i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.
The p-norm can be extended to vectors that have an infinite number of components, which yields the space . This contains as special cases:
- , the space of sequences whose series is absolutely convergent,
- , the space of square-summable sequences, which is a Hilbert space, and
- , the space of bounded sequences.
The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, for an infinite sequence of real (or complex) numbers, define the vector sum to be
while the scalar action is given by
Define the p-norm
Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, (1, 1, 1, …), will have an infinite p-norm (length) for every finite p ≥ 1. The space ℓp is then defined as the set of all infinite sequences of real (or complex) numbers such that the p-norm is finite.
One can check that as p increases, the set ℓp grows larger. For example, the sequence
is not in ℓ1, but it is in ℓp for p > 1, as the series
diverges for p = 1 (the harmonic series), but is convergent for p > 1.
One also defines the ∞-norm using the supremum:
and the corresponding space ℓ∞ of all bounded sequences. It turns out that[2]
if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider ℓp spaces for 1 ≤ p ≤ ∞.
The p-norm thus defined on ℓp is indeed a norm, and ℓp together with this norm is a Banach space. The fully general Lp space is obtained — as seen below — by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions. An integral instead of a sum is used to define the p-norm.
Lp spaces
Let 1 ≤ p < ∞ and (S, Σ, μ) be a measure space. Consider the set of all measurable functions from S to C (or R) whose absolute value raised to the p-th power has finite integral, or equivalently, that
The set of such functions forms a vector space, with the following natural operations:
for every scalar λ.
That the sum of two pth power integrable functions is again pth power integrable follows from the inequality |f + g|p ≤ 2p-1 (|f|p + |g|p). In fact, more is true. Minkowski's inequality says the triangle inequality holds for || · ||p. Thus the set of pth power integrable functions, together with the function || · ||p, is a seminormed vector space, which is denoted by .
<kernel>...</kernel>
This can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of || · ||p. Since for any measurable function f, we have that ||f||p = 0 if and only if f = 0 almost everywhere, the kernel of || · ||p does not depend upon p,
In the quotient space, two functions f and g are identified if f = g almost everywhere. The resulting normed vector space is, by definition,
For p = ∞, the space L∞(S, μ) is defined as follows. We start with the set of all measurable functions from S to C (or R) which are essentially bounded, i.e. bounded up to a set of measure zero. Again two such functions are identified if they are equal almost everywhere. Denote this set by L∞(S, μ). For f in L∞(S, μ), its essential supremum serves as an appropriate norm:
As before, we have
if f ∈ L∞(S, μ) ∩ Lq(S, μ) for some q < ∞.
For 1 ≤ p ≤ ∞, Lp(S, μ) is a Banach space. The fact that Lp is complete is often referred to as the Riesz-Fischer theorem. Completeness can be checked using the convergence theorems for Lebesgue integrals.
When the underlying measure space S is understood, Lp(S, μ) is often abbreviated Lp(μ), or just Lp. The above definitions generalize to Bochner spaces.
Special cases
When p = 2; like the ℓ2 space, the space L2 is the only Hilbert space of this class. In the complex case, the inner product on L2 is defined by
The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in L2 are sometimes called quadratically integrable functions, square-integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral Template:Harv.
If we use complex-valued functions, the space L∞ is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. An element of L∞ defines a bounded operator on any Lp space by multiplication.
The ℓp spaces (1 ≤ p ≤ ∞) are a special case of Lp spaces, when S is the set N of positive integers, and the measure μ is the counting measure on N. More generally, if one considers any set S with the counting measure, the resulting L p space is denoted ℓp(S). For example, the space ℓp(Z) is the space of all sequences indexed by the integers, and when defining the p-norm on such a space, one sums over all the integers. The space ℓp(n), where n is the set with n elements, is Rn with its p-norm as defined above. As any Hilbert space, every space L2 is linearly isometric to a suitable ℓ2(I), where the cardinality of the set I is the cardinality of an arbitrary Hilbertian basis for this particular L2.
Properties of Lp spaces
Dual spaces
The dual space (the space of all continuous linear functionals) of Lp(μ) for 1 < p < ∞ has a natural isomorphism with Lq(μ), where q is such that 1/p + 1/q = 1, which associates g ∈ Lq(μ) with the functional κp(g) ∈ Lp(μ)∗ defined by
The fact that κp(g) is well defined and continuous follows from Hölder's inequality. The mapping κp is a linear mapping from Lq(μ) into Lp(μ)∗, which is an isometry by the extremal case of Hölder's inequality. It is also possible to show (for example with the Radon–Nikodym theorem, see[3]) that any G ∈ Lp(μ)∗ can be expressed this way: i.e., that κp is onto. Since κp is onto and isometric, it is an isomorphism of Banach spaces. With this (isometric) isomorphism in mind, it is usual to say simply that Lq "is" the dual of Lp.
When 1 < p < ∞, the space Lp(μ) is reflexive. Let κp be the above map and let κq be the corresponding linear isometry from Lp(μ) onto Lq(μ)∗. The map
from Lp(μ) to Lp(μ)∗∗, obtained by composing κq with the transpose (or adjoint) of the inverse of κp, coincides with the canonical embedding J of Lp(μ) into its bidual. Moreover, the map jp is onto, as composition of two onto isometries, and this proves reflexivity.
If the measure μ on S is sigma-finite, then the dual of L1(μ) is isometrically isomorphic to L∞(μ) (more precisely, the map κ1 corresponding to p = 1 is an isometry from L∞(μ) onto L1(μ)∗).
The dual of L∞ is subtler. Elements of (L∞(μ))∗ can be identified with bounded signed finitely additive measures on S that are absolutely continuous with respect to μ. See ba space for more details. If we assume the axiom of choice, this space is much bigger than L1(μ) except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo-Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of ℓ∞ is ℓ1. [4]
Embeddings
Colloquially, if 1 ≤ p < q ≤ ∞, Lp(S, μ) contains functions that are more locally singular, while elements of Lq(S, μ) can be more spread out. Consider the Lebesgue measure on the half line (0, ∞). A continuous function in L1 might blow up near 0 but must decay sufficiently fast toward infinity. On the other hand, continuous functions in L∞ need not decay at all but no blow-up is allowed. The precise technical result is the following:
- Let 0 ≤ p < q ≤ ∞. Lq(S, μ) is contained in Lp(S, μ) iff S does not contain sets of arbitrarily large measure, and
- Let 0 ≤ p < q ≤ ∞. Lp(S, μ) is contained in Lq(S, μ) iff S does not contain sets of arbitrarily small non-zero measure.
In particular, if the domain S has finite measure, the bound (a consequence of Jensen's inequality)
means the space Lq is continuously embedded in Lp. That is to say, the identity operator is a bounded linear map from Lq to Lp. The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity I : Lq(S, μ) → Lp(S, μ) is precisely
the case of equality being achieved exactly when f = 1 a.e.[μ].
Dense subspaces
It is assumed that 1 ≤ p < ∞ throughout this section.
Let (S, Σ, μ) be a measure space. An integrable simple function f on S is one of the form
where aj is scalar and Aj ∈ Σ has finite measure, for j = 1, …, n. By construction of the integral, the vector space of integrable simple functions is dense in Lp(S, Σ, μ).
More can be said when S is a metrizable topological space and Σ its Borel σ–algebra, i.e., the smallest σ–algebra of subsets of S containing the open sets.
Suppose that V ⊂ S is an open set with μ(V) < ∞. It can be proved that for every Borel set A ∈ Σ contained in V, and for every ε > 0, there exist a closed set F and an open set U such that
It follows that there exists φ continuous on S such that
If S can be covered by an increasing sequence (Vn) of open sets that have finite measure, then the space of p–integrable continuous functions is dense in Lp(S, Σ, μ). More precisely, one can use bounded continuous functions that vanish outside one of the open sets Vn.
This applies in particular when S = Rd and when μ is the Lebesgue measure. The space of continuous and compactly supported functions is dense in Lp(Rd). Similarly, the space of integrable step functions is dense in Lp(Rd); this space is the linear span of indicator functions of bounded intervals when d = 1, of bounded rectangles when d = 2 and more generally of products of bounded intervals.
Several properties of general functions in Lp(Rd) are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on Lp(Rd), in the following sense: for every f ∈ Lp(Rd),
when t ∈ Rd tends to 0, where is the translated function defined by .
Applications
Lp spaces are widely used in mathematics and applications.
Hausdorff–Young inequality
The Fourier transform for the real line (resp. for periodic functions, cf. Fourier series) maps Lp(R) to Lq(R) (resp. Lp(T) to ℓq), where 1 ≤ p ≤ 2 and 1/p + 1/q = 1. This is a consequence of the Riesz-Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.
By contrast, if p > 2, the Fourier transform does not map into Lq.
Hilbert spaces
Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and ℓ2 are both Hilbert spaces. In fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ2(E), where E is a set with an appropriate cardinality.
Statistics
In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of Lp metrics, and measures of central tendency can be characterized as solutions to variational problems.
Lp for 0 < p < 1
Let (S, Σ, μ) be a measure space. If 0 < p < 1, then Lp(μ) can be defined as above: it is the vector space of those measurable functions f such that
As before, we may introduce the p-norm || f ||p = Np(f)1/p, but || · ||p does not satisfy the triangle inequality in this case, and defines only a quasi-norm. The inequality (a + b)p ≤ ap + bp, valid for a ≥ 0 and b ≥ 0 implies that Template:Harv
and so the function
is a metric on Lp(μ). The resulting metric space is complete; the verification is similar to the familiar case when p ≥ 1.
In this setting Lp satisfies a reverse Minkowski inequality, that is for u and v in Lp
This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces Lp for 1 < p < ∞ Template:Harv.
The space Lp for 0 < p < 1 is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is also locally bounded, much like the case p ≥ 1. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in ℓp or Lp([0, 1]), every open convex set containing the 0 function is unbounded for the p-quasi-norm; therefore, the 0 vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space S contains an infinite family of disjoint measurable sets of finite positive measure.
The only nonempty convex open set in Lp([0, 1]) is the entire space Template:Harv. As a particular consequence, there are no nonzero linear functionals on Lp([0, 1]): the dual space is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space Lp(μ) = ℓp), the bounded linear functionals on ℓp are exactly those that are bounded on ℓ1, namely those given by sequences in ℓ∞. Although ℓp does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.
The situation of having no linear functionals is highly undesirable for the purposes of doing analysis. In the case of the Lebesgue measure on Rn, rather than work with Lp for 0 < p < 1, it is common to work with the Hardy space Hp whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn–Banach theorem still fails in Hp for p < 1 Template:Harv.
L0, the space of measurable functions
The vector space of (equivalence classes of) measurable functions on (S, Σ, μ) is denoted L0(S, Σ, μ) Template:Harv. By definition, it contains all the Lp, and is equipped with the topology of convergence in measure. When μ is a probability measure (i.e., μ(S) = 1), this mode of convergence is named convergence in probability. The description is easier when μ is finite.
If μ is a finite measure on (S, Σ), the 0 function admits for the convergence in measure the following fundamental system of neighborhoods
The topology can be defined by any metric d of the form
where φ is bounded continuous concave and non-decreasing on [0, ∞), with φ(0) = 0 and φ(t) > 0 when t > 0 (for example, φ(t) = min(t, 1)). Such a metric is called Lévy-metric for L0. Under this metric the space L0 is complete (it is again an F-space). The space L0 is in general not locally bounded, and not locally convex.
For the infinite Lebesgue measure λ on Rn, the definition of the fundamental system of neighborhoods could be modified as follows
The resulting space L0(Rn, λ) coincides as topological vector space with L0(Rn, g(x) dλ(x)), for any positive λ–integrable density g.
Weak Lp
Let (S, Σ, μ) be a measure space, and f a measurable function with real or complex values on S. The distribution function of f is defined for t > 0 by
If f is in Lp(S, μ) for some p with 1 ≤ p < ∞, then by Markov's inequality,
A function f is said to be in the space weak Lp(S, μ), or Lp,w(S, μ), if there is a constant C > 0 such that, for all t > 0,
The best constant C for this inequality is the Lp,w-norm of f, and is denoted by
The weak Lp coincide with the Lorentz spaces Lp,∞, so this notation is also used to denote them.
The Lp,w-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for f in Lp(S, μ),
and in particular Lp(S, μ) ⊂ Lp,w(S, μ). Under the convention that two functions are equal if they are equal μ almost everywhere, then the spaces Lp,w are complete Template:Harv.
For any 0 < r < p the expression
is comparable to the Lp,w-norm. Further in the case p > 1, this expression defines a norm if r = 1. Hence for p > 1 the weak Lp spaces are Banach spaces Template:Harv.
A major result that uses the Lp,w-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.
Weighted Lp spaces
As before, consider a measure space (S, Σ, μ). Let be a measurable function. The w-weighted Lp space is defined as Lp(S, w dμ), where w dμ means the measure ν defined by
or, in terms of the Radon–Nikodym derivative,
The norm for Lp(S, w dμ) is explicitly
As Lp-spaces, the weighted spaces have nothing special, since Lp(S, w dμ) is equal to Lp(S, dν). But they are the natural framework for several results in harmonic analysis Template:Harv; they appear for example in the Muckenhoupt theorem: for 1 < p < ∞, the classical Hilbert transform is defined on Lp(T, λ) where T denotes the unit circle and λ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on Lp(Rn, λ). Muckenhoupt's theorem describes weights w such that the Hilbert transform remains bounded on Lp(T, w dλ) and the maximal operator on Lp(Rn, w dλ).
Lp spaces on manifolds
One may also define spaces on a manifold, called the intrinsic Lp spaces of the manifold, using densities.
See also
- Birnbaum–Orlicz space
- Hardy space
- Riesz–Thorin theorem
- Hölder mean
- Hölder space
- Root mean square
- Locally integrable function
- spaces over a locally compact group
- Minkowski distance
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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- Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
my web-site http://himerka.com/ - Template:Planetmath reference
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, page 16 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, Theorem 6.16 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 See Sections 14.77 and 27.44--47