Waring's problem

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

The weak and strong topologies

Let K be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications K will be either the field of complex numbers or the field of real numbers with the familiar topologies. Let X be a topological vector space over K. Namely, X is a K vector space equipped with a topology so that vector addition and scalar multiplication are continuous.

We may define a possibly different topology on X using the continuous (or topological) dual space X^*. The topological dual space consists of all linear functions from X into the base field K which are continuous with respect to the given topology. The weak topology on X is the initial topology with respect to X^*. In other words, it is the coarsest topology (the topology with the fewest open sets) such that each element of X^* is a continuous function. In order to distinguish the weak topology from the original topology on X, the original topology is often called the strong topology.

A subbase for the weak topology is the collection of sets of the form φ^-1(U) where φ ∈ X^* and U is an open subset of the base field K. In other words, a subset of X is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form φ^-1(U).

More generally, if F is a subset of the algebraic dual space, then the initial topology of X with respect to F, denoted by σ(X,F), is the weak topology with respect to F . If one takes F to be the whole continuous dual space of X, then the weak topology with respect to F coincides with the weak topology defined above.

If the field K has an absolute value $|\cdot |$, then the weak topology σ(X,F) is induced by the family of seminorms,

$\Vert x{\Vert }_f\stackrel{\text{def}}{=}|f(x)|$

for all f∈F and x∈X. In particular, weak topologies are locally convex. From this point of view, the weak topology is the coarsest polar topology; see weak topology (polar topology) for details. Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ(X,F) is precisely equal to F Template:Harv.

Weak convergence

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This is another simple approach for all property agents who wish to claim this PIC grant. The IRAS has some PIC seminar classes which might be specially created for all business homeowners. This seminar goes to share all important information about the PIC grant. You can even study on how you can apply this PIC scheme simply, legally, and rapidly. You don't have to spend your time for making use of this grant after taking this seminar. This seminar is officially held by the IRAS to coach all enterprise house owners in Singapore.

Ultimately, whose fault is it actually? I believe it is ultimately the federal government's fault, for not developing with higher legislature to control the trade, the individuals who work in the business and the developers. They need to change the law so that patrons pay their own agents and sellers pay their very own brokers to create a wholesome and honest enterprise atmosphere. Of course VicLand is also answerable for being so callous about not informing their own in home agents that unit was no longer out there. Most likely was not obtainable even once we went to view it yesterday. And ECG additionally has to take blame for being so unprofessional and virtually unethical by resorting to what equal to a black mail tactics to make a sale.

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Have accomplished at the very least 30 property transactions in Singapore previously 3 years, both as or on behalf of a Singapore licensed real estate agency; As an PERIOD Singapore property agent, I have entry to a database of Singapore actual estate listings together with new property for sale undertaking launches. I'm comfortable to work with you on another listings too. a. is resident in Singapore; a. he's resident in Singapore; Property firm that covers rental and sales in each the residential and business market in Singapore. RGF HR Agent Singapore Pte Ltd - Singapore Selling and renting property in Singapore. 6 Battery Highway, #32-01, Singapore 049909. We take care of with leasing, rental, sale and purchase of commercial and residential properties in Singapore. Downsides of investing in Singapore property

These are top 6 ways in which you can do to claim your PIC grant as knowledgeable property agent. There are some essential documents that you need to undergo the IRAS, so your software might be permitted rapidly. This PIC grant lets you develop your property agency company from the tax deduction or money payout choices. Her profession took her to Germany, U.Okay. and varied Asian international locations. Working with multi-nationwide corporations (e.g. Sony Ericsson) and schools (e.g. German School Singapore) made delicate in direction of the wants and emotions of of people of various ages and cultures, mixed with the ability to establish constructive relationships with households and other professionals. Southeast Asia Property Evaluation - MarketPulse Q1 2014 Browse Brokers by First Name

Making the leap to turning into a property investor We 'assumed' we'd just contact an estate agent and they'd have a list of properties for hire, if we didn't like what they had we would then transfer on to a different agent and keep doing this till we positioned an condominium that was suitable for us. Stamp obligation is to be paid inside 14 days from the date of acceptance of the OTP or Sale and Buy (S&P) Agreement. For extra data, please go to www.iras.gov.sg Estaland Property Associates Pte Ltd Completely different agents may bring the same buyer to view your house a second time, leading to confusion. Brokers who don't have exclusive rights to an inventory could also be much less keen to work for that itemizing. including Estate Agents Act and regulation of real property industry)

The weak topology is characterized by the following condition: a net (x_λ) in X converges in the weak topology to the element x of X if and only if φ(x_λ) converges to φ(x) in R or C for all φ in X* .

In particular, if x_n is a sequence in X, then x_n converges weakly to x if

$\varphi (x_n)\to \varphi (x)$

as n → ∞ for all φ ∈ X^*. In this case, it is customary to write

$x_n\stackrel{\mathrm{w}}{⟶}x$

or, sometimes,

$x_n\rightharpoonup x.$

Other properties

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

If X is a normed space, then the dual space X* is itself a normed vector space by using the norm ǁφǁ = sup_ǁxǁ≤1|φ(x)|. This norm gives rise to a topology, called the strong topology, on X*. This is the topology of uniform convergence. The uniform and strong topologies are generally different for other spaces of linear maps; see below.

The weak-* topology

A space X can be embedded into the double dual X** by

$x\mapsto T_x$

where

$T_x(\varphi )=\varphi (x).$

Thus T : X → X** is an injective linear mapping, though not necessarily surjective (spaces for which this canonical embedding is surjective are called reflexive). The weak-* topology on X* is the weak topology induced by the image of T: T(X) ⊂ X**. In other words, it is the coarsest topology such that the maps T_x, defined by T_x(φ) = φ(x) from X* to the base field R or C remain continuous.

Weak-* convergence

A net φ_λ in X* is convergent to φ in the weak-* topology if it converges pointwise:

${\varphi }_{\lambda }(x)\to \varphi (x)$

for all x in X. In particular, a sequence of φ_n ∈ X^* converges to φ provided that

${\varphi }_n(x)\to \varphi (x)$

for all x in X. In this case, one writes

${\varphi }_n\stackrel{w^{*}}{\to }\varphi$

as n → ∞.

Weak-* convergence is sometimes called the topology of simple convergence or the topology of pointwise convergence. Indeed, it coincides with the topology of pointwise convergence of linear functionals.

Other properties

By definition, the weak* topology is weaker than the weak topology on X*. An important fact about the weak* topology is the Banach–Alaoglu theorem: if X is normed, then the closed unit ball in X* is weak*-compact (more generally, the polar in X* of a neighborhood of 0 in X is weak*-compact). Moreover, the closed unit ball in a normed space X is compact in the weak topology if and only if X is reflexive.

In more generality, let F be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X*, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak-* topology.

If a normed space X is separable, then the weak-* topology is metrizable on the norm-bounded subsets of X*. It is not metrizable on all of X* unless the normed space X is countable-dimensional. If X is a Banach space, the weak-* topology is not metrizable on all of X* unless X is finite-dimensional.

Examples

Hilbert spaces

Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space L²(Rⁿ). Strong convergence of a sequence ψ_{k∈L²(Rⁿ) to an element ψ means that}

$\underset{{\mathbf{R}}^n}{\int }|{\psi }_k-\psi {|}^2\mathrm{d}\mu \to 0$

as k→∞. Here the notion of convergence corresponds to the norm on L².

In contrast weak convergence only demands that

$\underset{{\mathbf{R}}^n}{\int }{\overline{\psi }}_{k}f\mathrm{d}\mu \to \underset{{\mathbf{R}}^n}{\int }\overline{\psi }f\mathrm{d}\mu$

for all functions f∈L² (or, more typically, all f in a dense subset of L² such as a space of test functions, if the sequence {ψ_k} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in C.

For example, in the Hilbert space L²(0,π), the sequence of functions

${\psi }_k(x)=\sqrt{2/\pi }\sin (kx)$

form an orthonormal basis. In particular, the (strong) limit of ψ_k as k→∞ does not exist. On the other hand, by the Riemann–Lebesgue lemma, the weak limit exists and is zero.

Distributions

One normally obtains spaces of distributions by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on Rⁿ). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such as L². Thus one is led to consider the idea of a rigged Hilbert space.

Operator topologies

If X and Y are topological vector spaces, the space L(X,Y) of continuous linear operators f:X → Y may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space Y to define operator convergence Template:Harv. There are, in general, a vast array of possible operator topologies on L(X,Y), whose naming is not entirely intuitive.

For example, the strong operator topology on L(X,Y) is the topology of pointwise convergence. For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x∈X:

$f\mapsto \Vert f(x){\Vert }_Y.$

More generally, if a family of seminorms Q defines the topology on Y, then the seminorms p_q,x on L(X,Y) defining the strong topology are given by

$p_{q,x}:f\mapsto q(f(x)),$

indexed by q∈Q and x∈X.

In particular, see the weak operator topology and weak* operator topology.

See also

• Eberlein compactum, a compact set in the weak topology
• Weak convergence (Hilbert space)
• Weak-star operator topology
• Weak convergence of measures
• Topologies on the set of operators on a Hilbert space
• Vague topology

Notes

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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
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• 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