linear application of finite vector spaces are continuous. Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ two vector spaces of finite dimension where $V$ has dimension $m$ et $W$ dimension $n$. I'm trying to prove that if $T\in \mathcal L(V,W)$ then $T$ is continuous at $0$ (and thus continuous).
We have that there is $C>0$ s.t. $\|Tx\|_W\leq C\|x\|_V$ if and only if $\displaystyle\sup_{\|x\|_V=1}\|Tx\|_W<\infty $.
Let $B$ a basis of $V$ and $B'$ a basis of $W$. Then $$[Tx]_{B'}=[T]_{B'B}[x]_B,$$
and thus $$\sup_{\|[x]_B\|_{\mathbb R^m}=1}\|[Tx]_{B'}\|_{\mathbb R^n}=\sup_{\|[x]_B\|_{\mathbb R^m}=1}\|[T]_{B'B}[x]_B\|_{\mathbb R^n}.$$
I know that $A\longmapsto \displaystyle\sup_{\|x\|_{\mathbb R^m}=1}\|Ax\|_{\mathbb R^m}$ is a norm on the space of matrix and this is equivalent to the norm $A\longmapsto \sup_{i,j}|(A)_{ij}|$. Therefore $$\sup_{\|[x]_B\|_{\mathbb R^m}=1}\|[T]_{B'B}[x]_B\|_{\mathbb R^n}\leq C\sup_{i,j}|([T]_{B'B})_{ij}|<\infty.$$
Question / Problem : How can I prove that $$\sup_{\|x\|_V=1}\|Tx\|_W=\sup_{\|[x]_B\|_{\mathbb R^m}=1}\|[T]_{B'B}[x]_{B}\|_{\mathbb R^n} \ \ ?$$
By the way, which norm are $\|\cdot \|_{\mathbb R^n}$ and $\|\cdot\|_{\mathbb R^m}$ ? Are they the euclidienne norm ? 
 A: Lets take $\|\cdot \|_{\mathbb R^n}$ and $\|\cdot \|_{\mathbb R^m}$ the euclidian norm. Let $B'=\{w_1,...,w_n\}$ a basis of $W$. The application 
\begin{align*}
[\cdot ]_{B'}: (W,\|\cdot \|_W)&\longrightarrow (\mathbb R^n,\|\cdot \|_{\mathbb R^n})\\
 w&\longmapsto [w]_{B'}
\end{align*}
is a homeomorphism. The inverse is given by 
\begin{align*}
\operatorname{Real}_{B'}: (\mathbb R^n,\|\cdot \|_{\mathbb R^n})&\longrightarrow (W,\|\cdot \|_W)\\
\begin{pmatrix}x_1\\\vdots \\ x_n\end{pmatrix}&\longmapsto x_1w_1+\cdots +x_nw_n.
\end{align*}
The continuity is a consequence of the fact that all norm in finite dimension spaces are equivalent, and thus 
$$\|[x]_{B'}\|\leq C_1\max_{i=1,...,n}|x_i|\leq C_2\|x\|_W=C_2\|\operatorname{Real}_{B'}([x]_{B'})\|\leq C_3\max_{i=1,...,n}|x_i|$$$$\leq C_4 \|[x]_{B'}\|_{\mathbb R^n},$$
what gives the continuity of $[\cdot ]_{B'}$ and $\operatorname{Real}_{B'}$.
In particular $$\|Tx\|_W\leq D_1\|[Tx]_{B'}\|_{\mathbb R^n}=D_1\|[T]_{B'B}[x]_{B}\|_{\mathbb R^n}$$$$\leq D_1\|[T]_{B'B}\|_{\sigma }\|[x]_{B}\|_{\mathbb R^m}\underset{(*)}{\leq} D_2\|[T]_{B'B}\|_{\sigma }\|x\|_V,$$
where I denoted $$\|[T]_{B'B}\|_{\sigma }=\sup_{\|[x]_{B}\|_{\mathbb R^m}=1}\|[T]_{B'B}[x]_B\|,$$
and $(*)$ come from continuity of $[\cdot ]_{B}$. Finally,
$$\sup_{\|x\|_V\leq 1}\|Tx\|_W\leq D_2\|[T]_{B'B}\|_{\sigma }\leq D_3\sup_{i,j}|([T]_{B'B})_{ij}|<\infty .$$
