Prove that to every $A \in L(R^n,R^1)$ corresponds a unique $y \in R^n$ such that $Ax=\langle x, y\rangle$. Prove that $\|A\|=|y|$
Definition: For $A \in L(R^n,R^m)$, define the norm $\|A\|$ of A to be the sup of all numbers $|Ax|$, where x ranges over all vectors in $R^n$ with $|x| \leq 1$. The inequality $|Ax| \leq \|A\| \cdot |x|$ holds for all $x \in R^n$.
I am thinking about induction on the power of $n$, but not sure how to express each individual $A$, $x$ and $y$.