I am studying n-linear maps from Zorich, Mathematical analysis II, p. 49-53, where the author writes that "it is not difficult to prove that for mappings of finite-dimensional spaces the norm of a multilinear transformation is always finite". I am trying to do it myself, starting from the simple case of a linear $ A: X \to Y $ map between any two normed vector spaces $ X $ and $ Y $.
All I can write is:
$$||A||:=\sup_{|x|_X=1}|A(x)|_Y=\sup_{|x|_X=1}|A(x^1e_1+...+x^me_m)|_Y=\sup_{|x|_X=1}|x^1A(e_1)+...+x^mA(e_m)|_Y \leq \sup_{|x|_X=1} |x^1|\cdot |A(e_1)|_Y+...+|x^m|\cdot |A(e_m)|_Y\leq (|A(e_1)|_Y+...+|A(e_m)|_Y)\sup_{|x|_X=1} \max_{i=1,...,m} |x^i|$$
where $||\cdot||$ indicates the norm of a multilinear map, $|\cdot|_X$ and $|\cdot|_Y$ indicate the norm of a vector in space $X$ and $Y$, $(e_1,...,e_m)$ is a basis for $X$.
I can't conclude by finding a reason (in the general case we are considering) why $ \max_{i = 1, ..., m} | x^i | $ should be a limited quantity. Maybe this is a stupid question, but I can't find a general reason.