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

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

Examples

• $\mathbb{R}^n$ is a topological vector space

See also

• Topological group