Hilbert Spaces - Main Theorem

The Riesz Representation Theorem establishes a fundamental connection between continuous linear functionals and the inner product structure, characterizing the dual space of a Hilbert space.

TheoremRiesz Representation Theorem

Let $H$ be a Hilbert space. For every continuous linear functional $\phi : H \to \mathbb{K}$ , there exists a unique $y \in H$ such that $\phi(x) = \langle x, y \rangle$ for all $x \in H$ . Moreover, $\|\phi\| = \|y\|$ .

This theorem reveals that every continuous linear functional on a Hilbert space is represented by taking the inner product with some fixed vector. The correspondence $\phi \leftrightarrow y$ establishes an antilinear isometric isomorphism between $H^*$ (the dual space) and $H$ .

Proof

Existence: If $\phi = 0$ , take $y = 0$ . Otherwise, let $M = \ker(\phi)$ , which is a closed proper subspace. By the projection theorem, $H = M \oplus M^\perp$ , so there exists $z \in M^\perp$ with $z \neq 0$ .

For any $x \in H$ , consider $w = \phi(x)z - \phi(z)x$ . Then $\phi(w) = 0$ , so $w \in M$ . Since $z \perp M$ , we have $0 = \langle \phi(x)z - \phi(z)x, z \rangle = \phi(x)\|z\|^2 - \phi(z)\langle x, z \rangle$

Therefore, $\phi(x) = \langle x, z/\|z\|^2 \cdot \overline{\phi(z)} \rangle$ . Setting $y = z/\|z\|^2 \cdot \overline{\phi(z)}$ gives the representation.

Uniqueness: If $\langle x, y_1 \rangle = \langle x, y_2 \rangle$ for all $x$ , then $\langle x, y_1 - y_2 \rangle = 0$ for all $x$ . Taking $x = y_1 - y_2$ gives $\|y_1 - y_2\|^2 = 0$ , so $y_1 = y_2$ .

Norm: $|\phi(x)| = |\langle x, y \rangle| \leq \|x\| \|y\|$ by Cauchy-Schwarz, so $\|\phi\| \leq \|y\|$ . Taking $x = y$ gives $|\phi(y)| = \|y\|^2$ , so $\|\phi\| \geq \|y\|$ .

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ExampleApplications of Riesz Representation

Dual Space: The map $y \mapsto \phi_y$ where $\phi_y(x) = \langle x, y \rangle$ is an antilinear isometric isomorphism from $H$ to $H^*$
Weak Convergence: A sequence $(x_n)$ converges weakly to $x$ if $\langle x_n, y \rangle \to \langle x, y \rangle$ for all $y \in H$
Variational Problems: Solutions to $\min_{x \in H} \|x - x_0\|$ subject to $\phi(x) = c$ can be found using the representer $y$ of $\phi$

Remark

The Riesz Representation Theorem fails for general Banach spaces. The dual space $X^*$ of a Banach space $X$ is typically not isomorphic to $X$ itself, and there is no canonical way to represent functionals using vectors in $X$ . This special property of Hilbert spaces makes them particularly tractable.

This theorem is fundamental to the study of weak solutions of partial differential equations, optimization theory, and quantum mechanics, where states and observables are intimately connected through the inner product structure.