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

From Maths

## Statement

The *vector space* (considered with its usual topology) [ilmath]\mathbb{R}^n[/ilmath] is a topological vector space^{[1]}.

- That means the operations of:
- Addition, [ilmath]\mathcal{A}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n[/ilmath] given by [ilmath]\mathcal{A}:(u,v)\mapsto u+v[/ilmath] is continuous and
- Scalar multiplication, [ilmath]\mathcal{M}:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n[/ilmath] given by [ilmath]\mathcal{M}:(\lambda,v)\mapsto \lambda v[/ilmath] is also continuous

## Proof

Grade: C

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Advanced linear algebra - Roman - page 79. Should be easy enough to work out though once the topological basis stuff gets sorted

## References