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

Statement

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

• That means the operations of:
  1. 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
  2. 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