Suppose $V$ is a Hilbert space, $U$ is is a nonempty closed convex subset of $V$, and $g ∈ U$ is the unique element of $U$ with smallest norm. Prove that $\operatorname{Re}\langle g,h\rangle ≥ ‖‖^2$, for all $h ∈ U$.
I tried to use "convex": $tg+(1-t)h ∈ U$ for all $h ∈ U$, and $‖tg+(1-t)h‖ ≥ ‖‖$ as $g$ is the smallest. But cannot prove the inequality. Am I in the right way? Any help?
$\def\Re{\operatorname{Re}}$Here's an idea. Suppose that you have $\Re\langle h,g\rangle<\|g\|^2$ for some $h\in U$. Then, for $t\in(0,1)$ (the important thing is that $t>0$), \begin{align} \|tg+(1-t)h\|^2&=t^2\|g\|^2+(1-t)^2\|h\|^2+2t(1-t)\,\operatorname{Re}\langle g,h\rangle\\[0.3cm] &<t^2\|g\|^2+(1-t)^2\|h\|^2+2t(1-t)\,\|g\|^2\\[0.3cm] &\leq t^2\|g\|^2+(1-t)^2\|g\|^2+2t(1-t)\,\|g\|^2\\[0.3cm] &=\|g\|^2, \end{align} giving us a contradiction.