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] a topological vector space if it satisfies the following two properties:
  1. [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".
  2. [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]

Examples

• [ilmath]\mathbb{R}^n[/ilmath] is a topological vector space
• Example:A vector space that is not topological

See also

• Topological group