Linear mapping/Normed spaces/Finite-dimensional/Continuity/Fact

Let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be normed ${\displaystyle {}{\mathbb {K} }}$-vector spaces, and let

${\displaystyle \varphi \colon V\longrightarrow W}$

denote a linear mapping. Let ${\displaystyle {}V}$ be finite-dimensional.

Then ${\displaystyle {}\varphi }$ is continuous.