Difference between revisions of "Topological vector space"
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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}}. | ||
| + | ==Properties== | ||
| + | * [[For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal]] | ||
| + | ** Symbolically, if {{M|(X,\mathcal{J},\mathbb{K})}} be a TVS and let {{M|(Y,\mathbb{K})}} be a [[sub-vector space]] of {{M|X}} then: | ||
| + | *** {{M|(\exists U\in(\mathcal{J}-\{\emptyset\})[U\subseteq Y])\implies X\eq Y}} | ||
| + | |||
==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/> | ||
Latest revision as of 14:03, 16 February 2017
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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.
Definition
Let [ilmath](X,\mathbb{K})[/ilmath] be a vector space over the field of either the reals, so [ilmath]\mathbb{K}:\eq\mathbb{R} [/ilmath], or the complex numbers, so [ilmath]\mathbb{K}:\eq\mathbb{C} [/ilmath] and let [ilmath]\mathcal{J} [/ilmath] be a topology on [ilmath]X[/ilmath] so that [ilmath](X,\mathcal{ J })[/ilmath] is a topological space. We call the tuple:
- [ilmath](X,\mathcal{J},\mathbb{K})[/ilmath][Note 1] a topological vector space if it satisfies the following two properties[1][2]:
- [ilmath]\mathcal{A}:X\times X\rightarrow X[/ilmath] given by [ilmath]\mathcal{A}:(u,v)\mapsto u+v[/ilmath] is continuous - often said simply as "addition is continuous".
- [ilmath]\mathcal{M}:\mathbb{K}\times X\rightarrow X[/ilmath] given by [ilmath]\mathcal{M}:(\lambda, x)\mapsto \lambda x[/ilmath] 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 [ilmath]\mathbb{R} [/ilmath]'s topology being the same as the subspace topology of this) and the topology we imbue on [ilmath]X[/ilmath].
Properties
- For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal
- Symbolically, if [ilmath](X,\mathcal{J},\mathbb{K})[/ilmath] be a TVS and let [ilmath](Y,\mathbb{K})[/ilmath] be a sub-vector space of [ilmath]X[/ilmath] then:
- [ilmath](\exists U\in(\mathcal{J}-\{\emptyset\})[U\subseteq Y])\implies X\eq Y[/ilmath]
- Symbolically, if [ilmath](X,\mathcal{J},\mathbb{K})[/ilmath] be a TVS and let [ilmath](Y,\mathbb{K})[/ilmath] be a sub-vector space of [ilmath]X[/ilmath] then:
Examples
- [ilmath]\mathbb{R}^n[/ilmath] is a topological vector space
- Example:A vector space that is not topological
See also
Notes
- ↑ 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 [ilmath]X[/ilmath] than the field of a vector space is, so often we will just write:
- Let [ilmath](X,\mathbb{K})[/ilmath] be a topological vector space
References
- ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
- ↑ Advanced Linear Algebra - Steven Roman
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