Let $H$ be an euclidean Hilbert space, $K\subseteq H$ non-empty, closed, convex. Let $\beta$ be a continuous bilinear form satisfying $(\vert \beta(x,x)\vert\geq b\Vert x\Vert^2,\; x\in H)$ for some $b> 0$. In the end, I want to show that for every continuous linear form $\varphi\in H'$ there is some $x\in K$ such that $\left(\beta(x,y-x)\geq \varphi(y-x),\; y\in {\color{red}K}\right)$.
I got some help and showed that there is a continuous linear map $L\in\mathcal{L}(H)$ such that $\left(\beta(x,y)=\langle Lx,y\rangle,\; y\in H, x\in H\right)$. Now, I want to show that there is some $z\in H$ such that $$\beta(x,y-x)\geq \varphi(y-x)\Leftrightarrow \left(u=P_K(r(z-Lx)+x),\quad r>0\right)\; (\star)$$
I am told I could then use some fix-point theorem to get the result.
So, as the formulation of the problem was a bit messy, should the red $K$ in fact be $H$? How should $x$ be quantified in $(\star)$? I think it was meant that we can find a $z$ as above for every $x\in H$; does this make sense?
How can I show $(\star)$? What is $z$?