Difference between revisions of "Topological vector space"
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(Saving work, rewritten definition, 2 references. New reference allows us to use complex numbers rather than just reals.) |
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{{Stub page|msg=Find out if the space must be real, although it looks like it must be. Also find another reference. Demote to B once fleshed out.|grade=A*}} | {{Stub page|msg=Find out if the space must be real, although it looks like it must be. Also find another reference. Demote to B once fleshed out.|grade=A*}} | ||
==Definition== | ==Definition== | ||
| − | + | Let {{M|(X,\mathbb{K})}} be a [[vector space]] over the [[field]] of either [[the reals]], so {{M|\mathbb{K}:\eq\mathbb{R} }}, or [[the complex numbers]], so {{M|\mathbb{K}:\eq\mathbb{C} }} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.|X|J}} is a [[topological space]]. We call the [[tuple]]: | |
| − | # | + | * {{M|(X,\mathcal{J},\mathbb{K})}}<ref group="Note">This tuple doesn't really matter, nor does the order. I have done it this way for it topology first as in "topological vector space". The topology is "more implicit" when we speak of {{M|X}} than the field of a vector space is, so often we will just write: |
| − | # | + | * Let {{M|(X,\mathbb{K})}} be a topological vector space</ref> a ''topological vector space'' if it satisfies the following two properties{{rFAVIDMH}}{{rALASR}}: |
| + | *# {{M|\mathcal{A}:X\times X\rightarrow X}} given by {{M|\mathcal{A}:(u,v)\mapsto u+v}} is [[continuous]] - often said simply as "addition is continuous". | ||
| + | *# {{M|\mathcal{M}:\mathbb{K}\times X\rightarrow X}} given by {{M|\mathcal{M}:(\lambda, x)\mapsto \lambda x}} is also continuous, likewise also often said simply as "multiplication is continuous" | ||
| + | *#* {{Caveat|This is where the definition really matters}} as it relates [[the usual topology of the complex numbers]] (with {{M|\mathbb{R} }}'s topology being the same as the [[subspace topology]] of this) and the [[topology]] we imbue on {{M|X}}. | ||
==Examples== | ==Examples== | ||
* [[R^n is a topological vector space|{{M|\mathbb{R}^n}} is a topological vector space]] | * [[R^n is a topological vector space|{{M|\mathbb{R}^n}} is a topological vector space]] | ||
| + | * [[Example:A vector space that is not topological]] | ||
==See also== | ==See also== | ||
* [[Topological group]] | * [[Topological group]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
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Find out if the space must be real, although it looks like it must be. Also find another reference. Demote to B once fleshed out.
Contents
[hide]Definition
Let (X,K) be a vector space over the field of either the reals, so K:=R, or the complex numbers, so K:=C and let J be a topology on X so that (X,J) is a topological space. We call the tuple:
- (X,J,K)[Note 1] a topological vector space if it satisfies the following two properties[1][2]:
- A:X×X→X given by A:(u,v)↦u+v is continuous - often said simply as "addition is continuous".
- M:K×X→X given by M:(λ,x)↦λx is also continuous, likewise also often said simply as "multiplication is continuous"
- Caveat:This is where the definition really matters as it relates the usual topology of the complex numbers (with R's topology being the same as the subspace topology of this) and the topology we imbue on X.
Examples
See also
Notes
- Jump up ↑ This tuple doesn't really matter, nor does the order. I have done it this way for it topology first as in "topological vector space". The topology is "more implicit" when we speak of X than the field of a vector space is, so often we will just write:
- Let (X,K) be a topological vector space
References
- Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
- Jump up ↑ Advanced Linear Algebra - Steven Roman
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