Difference between revisions of "Topological vector space"
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*#* {{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}}. | *#* {{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== | ==Properties== | ||
| − | * [[For a vector subspace of a topological vector space if there exists | + | * [[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: | ** 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}[U\subseteq Y])\implies X\eq Y}} | + | *** {{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]] | ||
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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