Any linear transformation between two finite dimensional topological vector spaces is continuous. Theorem
Any linear transformation $T$ from one finite dimensional topological vector space $V$ into another finite dimensional topological vector $W$
space is necessarily continuous.
Unfortunately I don't be able to prove the statement so could someone prove it? Then if the statement is generally false is it false if $V=\Bbb R^m$ and $W=\Bbb R^m$ too? So could someone help me, please?
Let $T:V\longmapsto W$ a linear transformation, and $\mathcal{B}=\{e_1,\cdots,e_n\}$ a basis of V.
In finite dimension all norms are equivalents, let $||x||=\sum_{i=1}^n\vert x_i\vert$, then $$||f(x)||=||f(x_1e_1+\cdots+x_ne_n)|| \leqslant |x_1|||f(e_1)||+...+|x_n||f(e_n)||$$
Let $M=\underset{i\in\{1,...,n\}}{sup}||f(e_i)||$, then $$  |x_1|||f(e_1)||+...+|x_n||f(e_n)|| \leqslant M(|x_1|+...+|x_n|)=M||x|| $$
$\implies $ $$||f(x)|| \leqslant  M||x||$$
Let $T:V\longmapsto W$ a linear transformation, and $\mathcal{B}=\{e_1,\cdots,e_n\}$ a basis of V.
In finite dimension all norms are equivalents, let $||x||=\sum_{i=1}^n\vert x_i\vert$, then $$||f(x)||=||f(x_1e_1+\cdots+x_ne_n)|| \leqslant |x_1|||f(e_1)||+...+|x_n||f(e_n)||$$
Let $M=\underset{i\in\{1,...,n\}}{sup}||f(e_i)||$, then $$  |x_1|||f(e_1)||+...+|x_n||f(e_n)|| \leqslant M(|x_1|+...+|x_n|)=M||x|| $$
$\implies $ $$||f(x)|| \leqslant  M||x||$$