Functional Analysis_16. The Riesz Representation Theorem_16.1. Lax–Milgram theorem revisited

\begin{frame}[t]{16. The Riesz Representation Theorem \\ 16.1.Lax–Milgram theorem revisited} \pause Let $X$ be a Hilbert space and let $a : X \times X \to \mathbb{F}$ be a bounded bilinear form (if $\mathbb{F}=\mathbb{R}$), or a bounded sesquilinear form (if $\mathbb{F}=\mathbb{C}$). \pause Show that if there exist constants $C, \gamma greater than 0$ such that $$|a(x, y)| \leq C \|x\|_X \|y\|_X \quad \text{for all } x, y \in X,$$ $$\operatorname{Re} a(x, x) \geq \gamma \|x\|_X^2 \quad \text{for all } x \in X,$$ then there exists a unique operator $A : X \to X$ which is linear if $\mathbb{F}=\mathbb{R}$ and conjugate-linear if $\mathbb{F}=\mathbb{C}$, such that $$a(x, y) = \langle Ax, y \rangle \quad \text{for all } x, y \in X,$$ and $A$ is bounded. \pause Deduce from this that $A$ is invertible with $\|A^{-1}\|_{L(X)} \leq \gamma^{-1}$. \end{frame}