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In [[mathematics]], the '''Cayley transform''', named after [[Arthur Cayley]], has a cluster of related meanings. As originally described by {{Harvtxt|Cayley|1846}}, the Cayley transform is a mapping between [[skew-symmetric matrix|skew-symmetric matrices]] and [[special orthogonal matrix|special orthogonal matrices]]. In [[complex analysis]], the Cayley transform is a [[conformal map]]ping {{Harv|Rudin|1987}} in which the image of the upper complex half-plane is the unit disk {{Harv|Remmert|1991|pp=82ff, 275}}. And in the theory of [[Hilbert space]]s, the Cayley transform is a mapping between [[linear operator]]s {{Harv|Nikol’skii|2001}}.
In [[linear algebra]], the '''quotient''' of a [[vector space]] ''V'' by a [[linear subspace|subspace]] ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a '''quotient space''' and is denoted ''V''/''N'' (read ''V'' mod ''N'' or ''V'' by ''N'').


== Matrix map ==
== Definition ==
Among ''n''×''n'' [[square matrix|square matrices]] over the [[real number|reals]], with ''I'' the identity matrix, let ''A'' be any [[skew-symmetric matrix]] (so that ''A''<sup>T</sup>&nbsp;= −''A''). Then ''I''&nbsp;+&nbsp;''A'' is [[invertible matrix|invertible]], and the Cayley transform
Formally, the construction is as follows {{harv|Halmos|1974|loc=§21-22}}. Let ''V'' be a [[vector space]] over a [[field (mathematics)|field]] ''K'', and let ''N'' be a [[linear subspace|subspace]] of ''V''. We define an [[equivalence relation]] ~ on ''V'' by stating that ''x'' ~ ''y'' if ''x''&nbsp;&minus;&nbsp;''y'' &isin; ''N''. That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''.  From this definition, one can deduce that any element of ''N'' is related to the zero vector; in other words all the vectors in ''N'' get mapped into the equivalence class of the zero vector.


:<math> Q = (I - A)(I + A)^{-1} \,\!</math>
The [[equivalence class]] of ''x'' is often denoted
:[''x''] = ''x'' + ''N''
since it is given by
:[''x''] = {''x'' + ''n'' : ''n'' &isin; ''N''}.


produces an [[orthogonal matrix]], ''Q'' (so that ''Q''<sup>T</sup>''Q''&nbsp;= ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as <math> Q = (I + A)^{-1}(I - A)</math>. In fact, ''Q'' must have determinant +1, so is special orthogonal. Conversely, let ''Q'' be any orthogonal matrix which does not have −1 as an [[eigenvalue]]; then
The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equivalence classes over ''V'' by ~. Scalar multiplication and addition are defined on the equivalence classes by
*&alpha;[''x''] = [&alpha;''x''] for all &alpha; &isin; ''K'', and
*[''x'']&nbsp;+&nbsp;[''y''] = [''x''+''y''].
It is not hard to check that these operations are [[well-defined]] (i.e. do not depend on the choice of representative). These operations turn the quotient space ''V''/''N'' into a vector space over ''K'' with ''N'' being the zero class, [0].


:<math> A = (I - Q)(I + Q)^{-1} \,\!</math>
The mapping that associates to ''v''&nbsp;&isin;&nbsp;''V'' the equivalence class [''v''] is known as the '''quotient map'''.


is a skew-symmetric matrix. The condition on ''Q'' automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing ''Q''&nbsp;= ''A''<sup>c</sup> and ''A''&nbsp;= ''Q''<sup>c</sup>.
== Examples ==
Let ''X''&nbsp;=&nbsp;'''R'''<sup>2</sup> be the standard Cartesian plane, and let ''Y'' be a line through the origin in ''X''.  Then the quotient space ''X''/''Y'' can be identified with the space of all lines in ''X'' which are parallel to ''Y''.  That is to say that, the elements of the set ''X''/''Y'' are lines in ''X'' parallel to ''Y''.  This gives one way in which to visualize quotient spaces geometrically.


This version of the Cayley transform is its own functional inverse, so that ''A''&nbsp;= (''A''<sup>c</sup>)<sup>c</sup> and ''Q''&nbsp;= (''Q''<sup>c</sup>)<sup>c</sup>. A slightly different form is also seen {{Harv|Golub|Van Loan|1996}}, requiring different mappings in each direction (and dropping the superscript notation):
Another example is the quotient of '''R'''<sup>''n''</sup> by the subspace spanned by the first ''m'' standard basis vectors.  The space '''R'''<sup>''n''</sup> consists of all ''n''-tuples of real numbers (''x''<sub>1</sub>,…,''x''<sub>''n''</sub>).  The subspace, identified with '''R'''<sup>''m''</sup>, consists of all ''n''-tuples such that only the first ''m'' entries are non-zero: (''x''<sub>1</sub>,…,''x''<sub>''m''</sub>,0,0,…,0).  Two vectors of '''R'''<sup>''n''</sup> are in the same congruence class modulo the subspace if and only if they are identical in the last ''n''&minus;''m'' coordinates. The quotient space  '''R'''<sup>''n''</sup>/ '''R'''<sup>''m''</sup> is [[isomorphic]] to  '''R'''<sup>''n''&minus;''m''</sup> in an obvious manner.


:<math>\begin{align}
More generally, if ''V'' is an (internal) [[direct sum of vector spaces|direct sum]] of subspaces ''U'' and ''W'':
Q &{}= (I - A)^{-1}(I + A) \\
:<math>V=U\oplus W</math>
A &{}= (Q - I)(Q + I)^{-1}
then the quotient space ''V''/''U'' is naturally isomorphic to ''W'' {{harv|Halmos|1974|loc=Theorem 22.1}}.
\end{align}</math>


The mappings may also be written with the order of the factors reversed {{Harv|Courant|Hilbert|1989|loc=Ch.VII,&nbsp;&sect;7.2}}; however, ''A'' always commutes with (μ''I''&nbsp;±&nbsp;''A'')<sup>−1</sup>, so the reordering does not affect the definition.
An important example of a functional quotient space is a [[Lp_space#Lp_spaces|L<sup>p</sup> space]].


=== Examples ===
== Properties ==
In the 2×2 case, we have
:<math>
\begin{bmatrix} 0 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 0 \end{bmatrix}
\lrarr
\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} .
</math>
The 180° rotation matrix, −''I'', is excluded, though it is the limit as tan&nbsp;<sup>θ</sup>⁄<sub>2</sub> goes to infinity.


In the 3×3 case, we have
There is a natural [[epimorphism]] from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class [''x'']. The [[kernel (algebra)|kernel]] (or [[nullspace]]) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the [[short exact sequence]]
:<math>
:<math>0\to U\to V\to V/U\to 0.\,</math>
\begin{bmatrix} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{bmatrix}
\lrarr
\frac{1}{K}
\begin{bmatrix}
  w^2+x^2-y^2-z^2 & 2 (x y-w z) & 2 (w y+x z) \\
  2 (x y+w z) & w^2-x^2+y^2-z^2 & 2 (y z-w x) \\
  2 (x z-w y) & 2 (w x+y z) & w^2-x^2-y^2+z^2
\end{bmatrix} ,
</math>


where ''K''&nbsp;=&nbsp;''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>, and where ''w''&nbsp;=&nbsp;1. This we recognize as the rotation matrix corresponding to [[quaternion]]
If ''U'' is a subspace of ''V'', the [[dimension (vector space)|dimension]] of ''V''/''U'' is called the '''[[codimension]]''' of ''U'' in ''V''. Since a basis of ''V'' may be constructed from a basis ''A'' of ''U'' and a basis ''B'' of ''V''/''U'' by adding a representative of each element of ''B'' to ''A'', the dimension of ''V'' is the sum of the dimensions of ''U'' and ''V''/''U''. If ''V'' is [[finite-dimensional]], it follows that the codimension of ''U'' in ''V'' is the difference between the dimensions of ''V'' and ''U'' {{harv|Halmos|1974|loc=Theorem 22.2}}:
:<math>\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).</math>


:<math> w + \bold{i} x + \bold{j} y + \bold{k} z \,\!</math>
Let ''T'' : ''V'' &rarr; ''W'' be a [[linear operator]]. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' &isin; ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The [[first isomorphism theorem]] of linear algebra says that the quotient space ''V''/ker(''T'') is isomorphic to the image of ''V'' in ''W''. An immediate corollary, for finite-dimensional spaces, is the [[rank-nullity theorem]]: the dimension of ''V'' is equal to the dimension of the kernel (the ''nullity'' of ''T'') plus the dimension of the image (the ''rank'' of ''T'').


(by a formula Cayley had published the year before), except scaled so that ''w''&nbsp;= 1 instead of the usual scaling so that ''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>&nbsp;=&nbsp;1. Thus vector (''x'',''y'',''z'') is the unit axis of rotation scaled by tan&nbsp;<sup>θ</sup>⁄<sub>2</sub>. Again excluded are 180° rotations, which in this case are all ''Q'' which are [[symmetric matrix|symmetric]] (so that ''Q''<sup>T</sup>&nbsp;= ''Q'').
The [[cokernel]] of a linear operator ''T'' : ''V'' &rarr; ''W'' is defined to be the quotient space ''W''/im(''T'').


=== Other matrices ===
== Quotient of a Banach space by a subspace ==
We can extend the mapping to [[complex number|complex]] matrices by substituting "[[unitary matrix|unitary]]" for "orthogonal" and "[[skew-Hermitian matrix|skew-Hermitian]]" for "skew-symmetric", the difference being that the transpose (·<sup>T</sup>) is replaced by the [[conjugate transpose]] (·<sup>H</sup>). This is consistent with replacing the standard real [[inner product]] with the standard complex inner product. In fact, we may extend the definition further with choices of [[adjoint]] other than transpose or conjugate transpose.
If ''X'' is a [[Banach space]] and ''M'' is a [[closed set|closed]] subspace of ''X'', then the quotient ''X''/''M'' is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on ''X''/''M'' by
:<math> \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X. </math>
The quotient space ''X''/''M'' is [[complete space|complete]] with respect to the norm, so it is a Banach space.


Formally, the definition only requires some invertibility, so we can substitute for ''Q'' any matrix ''M'' whose eigenvalues do not include −1. For example, we have
=== Examples ===
:<math>
Let ''C''[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the [[sup norm]]. Denote the subspace of all functions ''f'' &isin; ''C''[0,1] with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space ''C''[0,1]&nbsp;/&nbsp;''M'' is isomorphic to '''R'''.
\begin{bmatrix} 0 & -a & ab - c \\ 0 & 0 & -b \\ 0 & 0 & 0 \end{bmatrix}
\lrarr
\begin{bmatrix} 1 & 2a & 2c \\ 0 & 1 & 2b \\ 0 & 0 & 1 \end{bmatrix} .
</math>
We remark that ''A'' is skew-symmetric (respectively, skew-Hermitian) if and only if ''Q'' is orthogonal (respectively, unitary) with no eigenvalue −1.
 
== Conformal map ==
[[Image:Cayley transform in complex plane.png|thumb|right| 300px|Cayley transform of upper complex half-plane to unit disk]]
In [[complex analysis]], the Cayley transform is a [[mapping (mathematics)|mapping]] of the [[complex plane]] to itself, given by
 
:<math> \operatorname{W} \colon z \mapsto \frac{z-\bold{i}}{z+\bold{i}} . </math>
 
This is a [[linear fractional transformation]], and can be extended to an [[automorphism]] of the [[Riemann sphere]] (the [[complex plane]] augmented with a point at infinity).  
 
Of particular note are the following facts:


* W maps the upper half plane of '''C''' [[conformal mapping|conformally]] onto the unit disc of '''C'''.
If ''X'' is a [[Hilbert space]], then the quotient space ''X''/''M'' is isomorphic to the [[Hilbert space#Orthogonal complements and projections|orthogonal complement]] of ''M''.
* W maps the real line '''R''' [[injective]]ly into the [[circle group|unit circle]] '''T''' (complex numbers of [[absolute value]] 1).  The image of '''R'''  is '''T''' with 1 removed.
* W maps the upper imaginary axis '''i''' <nowiki>[0, &infin;)</nowiki> [[bijection|bijectively]] onto the half-open interval <nowiki>[−1, +1)</nowiki>.
* W maps 0 to −1.
* W maps the point at infinity to 1.
* W maps −'''i''' to the point at infinity (so W has a [[pole (complex analysis)|pole]] at −'''i''').
* W maps −1 to '''i'''.
* W maps both <sup>1</sup>⁄<sub>2</sub>(−1&nbsp;+&nbsp;√3)(−1&nbsp;+&nbsp;'''i''') and <sup>1</sup>⁄<sub>2</sub>(1&nbsp;+&nbsp;√3)(1&nbsp;−&nbsp;'''i''') to themselves.


== Operator map ==
=== Generalization to locally convex spaces ===
An infinite-dimensional version of an [[inner product space]] is a [[Hilbert space]], and we can no longer speak of [[matrix (mathematics)|matrices]]. However, matrices are merely representations of [[linear operator]]s, and these we still have. So, generalizing both the matrix mapping and the complex plane mapping, we may define a Cayley transform of operators.
The quotient of a [[locally convex space]] by a closed subspace is again locally convex {{harv|Dieudonné|1970|loc=12.14.8}}. Indeed, suppose that ''X'' is locally convex so that the topology on ''X'' is generated by a family of [[seminorm]]s {''p''<sub>&alpha;</sub>|&alpha;&isin;''A''} where ''A'' is an index set.  Let ''M'' be a closed subspace, and define seminorms ''q''<sub>&alpha;</sub> by on ''X''/''M''
:<math>\begin{align}
U &{}= (A - \bold{i}I) (A + \bold{i}I)^{-1} \\
A &{}= \bold{i}(I + U) (I - U)^{-1}
\end{align}</math>
Here the domain of ''U'', dom&nbsp;''U'', is (''A''+'''i'''''I'')&nbsp;dom&nbsp;''A''. See [[self-adjoint operator#Extensions of symmetric operators|self-adjoint operator]] for further details.


== See also ==
:<math>q_\alpha([x]) = \inf_{x\in [x]} p_\alpha(x).</math>
* [[Bilinear transform]]


* [[Extensions of symmetric operators]]
Then ''X''/''M'' is a locally convex space, and the topology on it is the [[quotient topology]].


== References ==
If, furthermore, ''X'' is [[metrizable]], then so is ''X''/''M''.  If ''X'' is a [[Fréchet space]], then so is ''X''/''M'' {{harv|Dieudonné|1970|loc=12.11.3}}.


* {{Citation
==See also==
| last=Cayley
*[[quotient set]]
| first=Arthur
*[[quotient group]]
| author-link=Arthur Cayley
*[[quotient module]]
| year=1846
*[[quotient space]] (in [[topology]])
| title=Sur quelques propriétés des déterminants gauches
| journal=[[Journal für die reine und angewandte Mathematik]]
| volume=32
| pages=119–123
| url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D268141
| issn=0075-4102
}}; reprinted as article 52 (pp.&nbsp;332–336) in {{Citation
| last=Cayley
| first=Arthur
| author-link=Arthur Cayley
| year=1889
| title=The collected mathematical papers of Arthur Cayley
| publisher=[[Cambridge University Press]]
| volume=I (1841–1853)
| pages=332–336
| isbn=<!-- none given -->
| url=http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000349
}}
* {{Citation
  | last1=Courant
  | first1=Richard
  | author1-link=Richard Courant
  | last2=Hilbert
  | first2=David
  | author2-link=David Hilbert
  | title=Methods of Mathematical Physics
  | volume=1
  | edition=1st English
  | publisher=Wiley-Interscience
  | year=1989
  | place=New York
  | isbn=978-0-471-50447-4
}}
* {{Citation
  | last1=Golub
  | first1=Gene H.
  | author1-link=Gene H. Golub
  | last2=Van Loan
  | first2=Charles F.
  | author2-link=Charles F. Van Loan
  | title=Matrix Computations
  | edition=3rd
  | publisher=Johns Hopkins University Press
  | year=1996
  | place=Baltimore
  | isbn=978-0-8018-5414-9
}}
* {{Citation
| last=Nikol’skii
| first=N. K.
| contribution=Cayley transform
| contribution-url=http://www.encyclopediaofmath.org/index.php?title=Cayley_transform&oldid=12556
| title=[[Encyclopaedia of Mathematics]]
| year=2001
| publisher=[[Springer-Verlag]]
| isbn=978-1-4020-0609-8<!-- uncertain, web page gives 1402006098, which does not validate -->
}}; translated from the Russian {{Citation
| editor-last=Vinogradov
| editor-first=I. M.
| editor-link=Ivan Matveyevich Vinogradov
| title=Matematicheskaya Entsiklopediya
| place=Moscow
| publisher=Sovetskaya Entsiklopediya
| year=1977
}}
* {{Citation<!-- courtesy of [[User:CSTAR|]] -->
| last=Remmert
| first=Reinhold
| author-link=Reinhold Remmert
| translator=Robert B. Burckel (trans.)<!-- template does not provide for this -->
| title=Theory of Complex Functions
| series=Graduate Texts in Mathematics
| volume='''122''' of ''Graduate Texts in Mathematics'' (''Readings in Mathematics'')<!-- compensate for lack of template "series" support -->
| year=1991
| publisher=[[Springer-Verlag]]
| place=New York
| isbn=978-0-387-97195-7
}}, translated by Robert B. Burckel from {{Citation
| unused_data=Grundwissen Mathematik 5
| last=Remmert
| first=Reinhold
| author-link=Reinhold Remmert
| title=Funktionentheorie I
| edition=2nd
| year=1989
| publisher=[[Springer-Verlag]]
| isbn=978-3-540-51238-7
}}
* {{Citation
| last=Rudin
| first=Walter
| author-link=Walter Rudin
| title=Real and Complex Analysis
| edition=3rd<!-- date was 1966, no edition given -->
| publisher=McGraw-Hill
| year=1987<!-- March 1 -->
| isbn=978-0-07-100276-9
}}


== External links ==
==References==
* {{PlanetMath
* {{citation|first=Paul|last=Halmos|authorlink=Paul Halmos|title=Finite dimensional vector spaces|publisher=Springer|year=1974|isbn=978-0-387-90093-3}}.
| urlname=CayleyTransform
* {{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on analysis, Volume II|publisher=Academic Press|year=1970}}.
| title=Cayley's parameterization of orthogonal matrices
| id=6535
}}


[[Category:Conformal mapping]]
[[Category:Linear algebra]]
[[Category:Transforms]]
[[Category:Functional analysis]]


[[fr:Transformation de Cayley]]
[[ca:Espai vectorial quocient]]
[[it:Trasformata di Cayley]]
[[de:Faktorraum]]
[[ru:Преобразование Мёбиуса#Примеры]]
[[it:Spazio vettoriale quoziente]]
[[uk:Перетворення Келі]]
[[he:מרחב מנה (אלגברה לינארית)]]
[[ja:商線型空間]]
[[pl:Przestrzeń ilorazowa (algebra liniowa)]]
[[ru:Факторпространство по подпространству]]
[[zh:商空间 (线性代数)]]

Revision as of 22:43, 12 August 2014

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).

Definition

Formally, the construction is as follows Template:Harv. Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector; in other words all the vectors in N get mapped into the equivalence class of the zero vector.

The equivalence class of x is often denoted

[x] = x + N

since it is given by

[x] = {x + n : n ∈ N}.

The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by

  • α[x] = [αx] for all α ∈ K, and
  • [x] + [y] = [x+y].

It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0].

The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map.

Examples

Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. This gives one way in which to visualize quotient spaces geometrically.

Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1,…,xn). The subspace, identified with Rm, consists of all n-tuples such that only the first m entries are non-zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic to Rn−m in an obvious manner.

More generally, if V is an (internal) direct sum of subspaces U and W:

V=U⊕W

then the quotient space V/U is naturally isomorphic to W Template:Harv.

An important example of a functional quotient space is a Lp space.

Properties

There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence

0→U→V→V/U→0.

If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U Template:Harv:

codim(U)=dim⁡(V/U)=dim⁡(V)−dim⁡(U).

Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x ∈ V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem of linear algebra says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank-nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).

The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).

Quotient of a Banach space by a subspace

If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M by

‖[x]‖X/M=infm∈M‖x−m‖X.

The quotient space X/M is complete with respect to the norm, so it is a Banach space.

Examples

Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R.

If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.

Generalization to locally convex spaces

The quotient of a locally convex space by a closed subspace is again locally convex Template:Harv. Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα|α∈A} where A is an index set. Let M be a closed subspace, and define seminorms qα by on X/M

qα([x])=infx∈[x]pα(x).

Then X/M is a locally convex space, and the topology on it is the quotient topology.

If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M Template:Harv.

See also

References

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    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.

ca:Espai vectorial quocient de:Faktorraum it:Spazio vettoriale quoziente he:מרחב מנה (אלגברה לינארית) ja:商線型空間 pl:Przestrzeń ilorazowa (algebra liniowa) ru:Факторпространство по подпространству zh:商空间 (线性代数)

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