Please help me understand why a particular statement follows from this lemma.
Lemma: Let $K$ be a convex subset of Hilbert Space $H$ and $p$ $\in$ $H$. Suppose that $q$ $\in$ $K$ is such that dist($p$,$K$)$=$$\parallel$$p$$-$$q$$\parallel$ (where dist($p$,$K$) $:=$ inf$\parallel$$p$$-$$q$$\parallel$, $q$ $\in$ $K$). Then Re$\langle$$p-q,q'-q$$\rangle$$\leq$$0$ $\forall$$q'$$\in$$K$
Statement: In the case that $K$ is a linear subspace of $H$, then Re$\langle$$p-q,q'-q$$\rangle$$\leq$$0$ $\forall$$q'$$\in$$K$ $\Leftrightarrow$ $\langle$$p-q,q'$$\rangle$ $=$ $0$ $\forall$$q'$$\in$$K$
Best Answer
Hint: Let $y:=p-q$. For all $x\in K$ we have $$Re\langle y,x\rangle, \ Re\langle y, -x\rangle, \ Re\langle y, ix\rangle, \ Re\langle y, -ix\rangle \ \le 0\,.$$