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

Proof

The image under the mapping is also a finite-dimensional vector space; therefore, we may assume that both spaces are finite-dimensional. We also may assume that ${}V=\mathbb {R} ^{n}$ and ${}W=\mathbb {R} ^{m}$. Because of fact, we may assume that we have the maximum norm on both sides. Let ${}a$ be the maximum of the absolute values of the entries in the describing matrix ${}M={\left(a_{ij}\right)}_{ij}$ of ${}\varphi$, with respect to the standard bases. For ${}v\in \mathbb {R} ^{n}$ with

${}\Vert {v}\Vert _{\rm {max}}\leq 1\,,$

we obtain

${}\Vert {\varphi (v)}\Vert _{\rm {max}}=\Vert {\begin{pmatrix}\sum _{j=1}^{n}a_{1j}v_{j}\\\vdots \\\sum _{j=1}^{n}a_{nj}v_{j}\end{pmatrix}}\Vert _{\rm {max}}={\max {\left(\vert {\sum _{j=1}^{n}a_{ij}v_{j}}\vert ,i=1,\ldots ,m\right)}}\leq na\,.$

Therefore, continuity follows from fact.

To fact