This is the Stampacchia theorem given in Brezis' Functional Analysis: (here the angular brackets mean the pairing between $H$ and its dual)
Theorem 5.6 (Stampacchia) Let $H$ be a Hilbert space over $\mathbb H$. Assume that $a(u,v)$ is a continuous coercive bilinear form. Let $K$ be a closed convex subset of $H$. Then for any $\varphi\in H^*$, there exists a unique element $u\in K$ such that $$a(u,v-u)\geq\langle\varphi,v-u\rangle,\ \forall v\in K$$
I am trying to understand it from a geometric or algebraic view. For example, its corollary when $K=H$:
Corollary 5.8 (Lax-Milgram) Let $H, a(u,v), \varphi$ be as above. Then there exists a unique $u\in H$ such that $$a(u,v)=\langle\varphi,v\rangle,\ \forall v\in H$$
can be interpreted as: any bounded linear functional $\varphi\in H^*$ can be represented by $a(u,-)$ for some suitable, uniquely determined $u$. More succinctly $H=H^*$ by $u\mapsto a(u,-)$. Now this is somewhat like Riesz's representation theorem (although this map may not be an isometry), and is easy to understand and remember.
Question: Is there a more understandable interpretation of Stampacchia's theorem, geometric or algebraic, that makes it easier to remember or provides intuition for its motivation?