Topological vector space
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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 nonempty 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 subvector 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 subvector 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

