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

Statement

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

• That means the operations of:
  1. Addition, $\mathcal{A}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ given by $\mathcal{A}:(u,v)\mapsto u+v$ is continuous and
  2. Scalar multiplication, $\mathcal{M}:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ given by $\mathcal{M}:(\lambda,v)\mapsto \lambda v$ 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