Let $V$ be an inner product space, and $v\in V$. I want to show that $$||v||=\sup\{\langle\,v,w\rangle:||w||=1\}$$
What I know: Since this is a generic inner product, $||v||=\sqrt{\langle\,v,v\rangle}$. So, if $||v||=0$ then $v=0$. The equality is trivial if $v=0$, so I know I need to consider when $v\neq 0$. Also, we have that $||w||=1$ this implies that $w=\frac{v}{||v||}$ if we use it in terms of $v$.
I am not sure if I am pulling the correct information, but I am confused on how to proceed to prove the equality statement.