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

A tuple, [ilmath](V,\mathcal{J})[/ilmath] where [ilmath]V[/ilmath] is a real vector space (a vector space over the field of real numbers, [ilmath]\mathbb{R} [/ilmath]) and [ilmath]\mathcal{J} [/ilmath] is a topology on the set [ilmath]V[/ilmath] is called a topological vector space if:

1. The operation of addition is continuous, that is to say that the map [ilmath]\mathcal{A}:V\times V\rightarrow V[/ilmath] given by [ilmath]\mathcal{A}:(u,v)\mapsto u+v[/ilmath] is continuous
2. The operation of scalar multiplication is continuous, that is the map [ilmath]\mathcal{M}:\mathbb{R}\times V\rightarrow V[/ilmath] by [ilmath]\mathcal{M}:(\lambda,v)\mapsto \lambda v[/ilmath] is also continuous

Examples

• [ilmath]\mathbb{R}^n[/ilmath] is a topological vector space

See also

• Topological group