Normed finite-dimensional vector spaces/Linear mapping/Norm estimate/Fact/Proof/Exercise

Let ${}V$ and ${}W$ be finite-dimensional normed ${}{\mathbb {K} }$-vector spaces, and let ${}\varphi \colon V\rightarrow W$ denote a linear mapping. Show the estimate

${}\Vert {\varphi (v)}\Vert \leq \Vert {\varphi }\Vert \cdot \Vert {v}\Vert \,$

for all ${}v\in V$.

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