Riesz Representation Theorem

• categories: Functional analysis, Theorem

The Riesz Representation Theorem is a foundational result in functional analysis, particularly in Hilbert Space theory. It establishes a correspondence between linear functionals and elements of a Hilbert space, providing a way to “represent” each continuous linear functional as an inner product with a unique vector in the space.

Statement

Let $H$ be a Hilbert space over $\mathbb{R}$ or $\mathbb{C}$. For every continuous linear functional $f:H\to \mathbb{F}$ (where $\mathbb{F}$ is $\mathbb{R}$ or $\mathbb{C}$), there exists a unique vector $u\in H$ such that for all $v\in H$,

$$f(v)=⟨v,u⟩.$$

The vector $u$ is uniquely determined and satisfies $\Vert f\Vert =\Vert u\Vert$, where $\Vert f\Vert$ denotes the operator norm of the functional $f$.

Intuition

The theorem tells us that every continuous linear functional on a Hilbert space $H$ can be represented as taking the inner product with a fixed vector in $H$. In other words, functionals in $H^{\ast }$ (the dual space of $H$) can be identified with elements of $H$ itself, allowing us to treat $H$ and its dual space $H^{\ast }$ as isometrically isomorphic.

Properties and Implications

1. Isomorphism Between $H$ and $H^{\ast }$: The map $\Phi :H\to H^{\ast }$ given by $\Phi (u)(v)=⟨v,u⟩$ is an isometric isomorphism. Thus, $H$ and $H^{\ast }$ are essentially the same in the sense of normed spaces.

2. Norm Preservation: The norm of the functional $f$ equals the norm of its corresponding vector $u$, i.e., $\Vert f\Vert =\Vert u\Vert$.

3. Self-Duality: In a Hilbert space, this theorem allows us to “represent” every continuous linear functional as an element of the space itself, unlike in general Banach spaces.

Applications

• Solution to Optimization Problems: This theorem provides the foundation for various optimization techniques, such as finding the best approximation in least squares problems.
• Weak Convergence: It allows for an inner product-based characterization of weak convergence in Hilbert spaces.
• Spectral Theory: The theorem is instrumental in developing the spectral theory of operators on Hilbert spaces by connecting functionals and vectors.

Proof Sketch

1. Define Candidate Vector: For a given $f\in H^{\ast }$, if $f=0$, let $u=0$. Otherwise, use the Hilbert space structure to find a vector $u$ such that $f(v)=⟨v,u⟩$ for all $v\in H$.

2. Uniqueness and Norm Preservation: Uniqueness of $u$ follows from the properties of inner products, and norm preservation is verified using theCauchy–Schwarz Inequality and properties of $f$.

The Riesz Representation Theorem elegantly bridges the structure of a Hilbert space and its dual, making it one of the most powerful tools in functional analysis.