On page 175 of Edward's Advanced Calculus of Several Variables, there is the following theorem:
Theorem 2.3 Let $A = (a_{ij})$ be the matrix of the linear mapping $L:\mathbb{R^n} \rightarrow \mathbb{R^m}$, that is, $Lx = Ax$ for all $x \in \mathbb{R^n}.$ Then $$||L|| = ||A||.$$
Now, there is a rather long proof for this theorem in the book. I was wondering if I could prove it by saying that the result follows immediately from Riesz Representation Theorem. If not, how can I prove this theorem by using it? I have the intuition that it is easier through that path.
Note: The proof written in the book uses propositions on the sup norm and a lot of inequalities.