supremum on quadratic and bilinear form

Question

Let $V$ be a real normed vector space. Denote by $V_1$ the subset of $V$ of vectors with norm $\le 1$. Consider a positive-semidefinite symmetric bilinear form $\langle \cdot, \cdot\rangle : V\times V\to \Bbb{R}$, not necessarily the one giving the norm.

Is it true, in general, that $$ \sup_{v,w\in V_1} \langle v, w \rangle = \sup_{v\in V_1} \langle v, v \rangle $$ or do we just have an inequality?

Edit: clarified a bit.

Answer 1

Since $\{(v,v) | v\in V_1 \}\subset \{(v,w) | v\in V_1 w \in V_1 \} $ you get the one direction. The other direction is Cauchy Schwarz.

Edit: $\langle v,w\rangle\leq \sqrt{\left\|v\right\|\left\|w\right\|}\leq \sqrt{\sup_{v\in V_1}\left\|v\right\|^2}=\sup_{v\in V_1}\sqrt{\left\|v\right\|^2}=\sup_{v\in V_1}\left\|v\right\|$ where I used that the square root is monotonic to get the supremum out of it.