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In [[mathematics]], the '''Kuratowski embedding''' allows one to view any [[metric space]] as a subset of some [[Banach space]]. It is named after [[Kazimierz Kuratowski]]. | |||
Specifically, if (''X'',''d'') is a metric space, ''x''<sub>0</sub> is a point in ''X'', and ''C<sub>b</sub>''(''X'') denotes the Banach space of all bounded [[continuous function|continuous]] real valued functions on ''X'' with the [[uniform norm|supremum norm]], then the map | |||
:<math>\Phi : X \rarr C_b(X)</math> | |||
defined by | |||
:<math>\Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox{for all}\quad x,y\in X</math> | |||
is an [[isometry]].<ref>{{citation|title=Geometric embeddings of metric spaces|url=http://www.math.jyu.fi/research/reports/rep90.ps|author=Juha Heinonen|date=January 2003|accessdate=6 January 2009}}</ref> | |||
Note that this embedding depends on the chosen point ''x''<sub>0</sub> and is therefore not entirely canonical. | |||
The '''Kuratowski–Wojdysławski''' theorem states that every bounded metric space ''X'' is isometric to a [[closed set|closed subset]] of a [[Convex set|convex]] subset of some Banach space.<ref>{{citation|author=[[Karol Borsuk]]|title=Theory of retracts|year=1967|place=Warsaw}}. Theorem III.8.1</ref> (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry | |||
:<math>\Psi : X \rarr C_b(X)</math> | |||
defined by | |||
:<math>\Psi(x)(y) = d(x,y) \quad\mbox{for all}\quad x,y\in X</math> | |||
The convex set mentioned above is the [[convex hull]] of Ψ(''X''). | |||
In both of these embedding theorems, we may replace ''C<sub>b</sub>''(''X'') by the Banach space ''ℓ''<sup> ∞</sup>(''X'') of all bounded functions ''X'' → '''R''', again with the supremum norm, since ''C<sub>b</sub>''(''X'') is a closed linear subspace of ''ℓ''<sup> ∞</sup>(''X''). | |||
These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are [[vector space]]s which allows one to add points and do elementary geometry involving lines and planes etc.; and they are [[complete space|complete]]. Given a function with [[codomain]] ''X'', it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing ''X''. | |||
== History == | |||
Formally speaking, this embedding was first introduced by [[Kazimierz Kuratowski|Kuratowski]],<ref>Kuratowski, C. (1935) "Quelques problèmes concernant les espaces métriques non-separables", [[Fundamenta Mathematica]] 25: 534-545.</ref> | |||
but a very close variation of this embedding appears already in the paper of Fréchet<ref>''Fréchet M.'' (1906) "Sur quelques points du calcul fonctionnel", [[Rendiconti del Circolo Matematico di Palermo]] 22: 1—74.</ref> where he first introduces the notion of metric space. | |||
==See also== | |||
*[[Tight span]], an embedding of any metric space into an [[injective metric space]] defined similarly to the Kuratowski embedding | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Kuratowski Embedding}} | |||
[[Category:Functional analysis]] | |||
[[Category:Metric geometry]] | |||
Latest revision as of 00:32, 20 March 2013
In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski.
Specifically, if (X,d) is a metric space, x0 is a point in X, and Cb(X) denotes the Banach space of all bounded continuous real valued functions on X with the supremum norm, then the map
defined by
Note that this embedding depends on the chosen point x0 and is therefore not entirely canonical.
The Kuratowski–Wojdysławski theorem states that every bounded metric space X is isometric to a closed subset of a convex subset of some Banach space.[2] (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry
defined by
The convex set mentioned above is the convex hull of Ψ(X).
In both of these embedding theorems, we may replace Cb(X) by the Banach space ℓ ∞(X) of all bounded functions X → R, again with the supremum norm, since Cb(X) is a closed linear subspace of ℓ ∞(X).
These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are vector spaces which allows one to add points and do elementary geometry involving lines and planes etc.; and they are complete. Given a function with codomain X, it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing X.
History
Formally speaking, this embedding was first introduced by Kuratowski,[3] but a very close variation of this embedding appears already in the paper of Fréchet[4] where he first introduces the notion of metric space.
See also
- Tight span, an embedding of any metric space into an injective metric space defined similarly to the Kuratowski embedding
References
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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 - ↑ 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 III.8.1 - ↑ Kuratowski, C. (1935) "Quelques problèmes concernant les espaces métriques non-separables", Fundamenta Mathematica 25: 534-545.
- ↑ Fréchet M. (1906) "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo 22: 1—74.