View source for Topological vector space

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==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==
* [[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==
* [[Topological group]]
==Notes==
<references group="Note"/>
==References==
<references/>
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