Lax-Milgram's theorem states that for a continuous bilinear form $a: H \times H \to \mathbb{R}$ with $$|a(x,y)| \leq \Lambda \|x \|_H \| y \|_H$$ and for some $\lambda > 0$ $$a(x,x) \geq \lambda \| x \|_H^2$$ there is a continuous bijection $A \in L(H)$ with $$a(x,y) = \langle Ax, y \rangle_H$$ and $\| A \|_{L(H)} \leq \Lambda$, $\| A^{-1} \|_{L(H)} \leq \lambda^{-1}$.
To prove this we apply Riesz' representation theorem on $\ell_x: y \mapsto a(x,y)$. This gives us the existance of $Ax = J^{-1} \ell_x \in H$, where $J$ is the isometric isomorphism given by Riesz' theorem. However, isn't $J…