Difference between revisions of "R^n is a topological vector space"

Latest revision as of 14:25, 16 August 2016

Statement

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

• 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

References

1. Jump up ↑ Advanced Linear Algebra - Steven Roman