Theorem 12.6.14 (Riesz Representation Theorem).label Let $H$ be a Hilbert space over $K \in \RC$. For each $x \in H$, let
\[\phi_{x}: H \to \complex \quad \phi_{x}(y) = \dpn{y, x}{E}\]
then the mapping $H \to H^{*}$ defined by $x \mapsto \phi_{x}$ is an isometric conjugate linear isomorphism.
Proof. By the Cauchy-Schwarz inequality and definition of the norm, $x \mapsto \phi_{x}$ is an isometric conjugate linear map.
Let $\phi \in H^{*}$, then $\ker \phi$ is a closed subspace of codimension one. By Theorem 12.6.12, $(\ker \phi)^{\perp} \ne \emptyset$. Let $y \in (\ker \phi)^{\perp}$ with $\norm{y}_{H} = 1$, then for any $x \in H$,
\begin{align*}\phi(x)&= \phi(P_{\ker \phi}x) + \phi(P_{(\ker \phi)^\perp}x) \\&= \dpn{P_{\ker \phi}x, \phi(y)y}{H}+ \dpn{P_{(\ker \phi)^\perp}x, \phi(y)y}{H}= \dpn{x, \phi(y)y}{H}\end{align*}
$\square$
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