I am reading Brezis book "FA, Sobolev Sp. and PDEs" and I am working through the proof of Stampacchia theorem (5.6 page 138 2010 edition) and I am particulary interested in Lax-Milgram theorem (which is given as a Corollary 5.8).
Lax-Milgram theorem is proven as a corollary of Stampacchia theorem invoking corollary 5.4, which states
Suppose $W$ is a closed linear subspace of $H.$ For $x\in H,$ $y=P_Kx$ is characterized by the property that for all $\omega \in W$ $$ y\in W \ \text{and} \ \langle x-y, \omega \rangle =0.$$
How does one prove Lax Milgram from this?
My explanation
What I would say is that taking $H=K$ we argue as in Stampacchia theorem (keeping in mind that for $H$ the above argument applies with $H=W$) to obtain the unique $u \in H$ such that for all $v \in H$ $$a(u,v-u+u)=\ell (v-u+u),$$ where we have an equality ($= 0$) instead of an inequality ($\leq 0$) precisely because we have so in the corollary 5.4, and the same goes about the $+u$ factor. And then when $a$ is symmetric the Stampacchia's argument completely carries over without any change so that the minimizing function is the same but we minimize it over $H$ instead of $K.$
Is it correct? I would like to make it more precise, any suggestions?
Note I found this recent question which asks the same thing but has received answers that are not very to the point, since they just give standard proofs of Lax-Milgram without any reference to Stampacchia's theorem, which is not what the other (and my own) question was about at all. So I ask hoping to receive a more on-topic answer.
Stampacchia theorem yields the existence of $u$ such that $$ a(u,v-u) \ge l(v-u) \quad\forall v\in H. $$ Take $w\in H$. Then we can set $v = \pm w+u$ to obtain the equality $$ a(u,w) = l(w) \quad \forall w\in H. $$
If $a$ is symmetric then $$ (u,v) \mapsto a(u,v) $$ is an inner product that induces a norm that is equivalent to the norm on $V$. And the energy minimization problem is a quadratic minimization problem.
To prove the claim about the energy minimization, the book says 'argue as in Corollary 5.4' (and not apply Corollary 5.4). Corollary 5.4 makes a connection between a projection problem (which is the minimization of a quadratic function like our energy minimization) and an inequality (which in our case is the desired equation). So to prove the equation you have to repeat the arguments in the proof of Corollary and apply them to the new situation. A more direct proof can be found in my answer to the other question.