Riesz Representation Theorem

The Riesz Representation Theorem serves as a key result in functional analysis, establishing a connection between linear functionals and measures or inner products, depending on the context. The theorem has multiple versions, and two notable ones are as follows:

Riesz Representation Theorem for Hilbert Spaces

Let $H$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$, and let $f: H \rightarrow \mathbb{C}$ (or $f: H \rightarrow \mathbb{R}$) be a continuous linear functional. Then there exists a unique vector $y \in H$ such that for every $x \in H$ $$f(x) = \langle x, y \rangle$$

Here, $\langle \cdot, \cdot \rangle$ denotes the inner product in $H$.

Riesz Representation Theorem for Measures

Let $X$ be a locally compact Hausdorff space, and let $C_c(X)$ be the space of all continuous functions $f: X \rightarrow \mathbb{C}$ (or $\mathbb{R}$) with compact support. If $\Lambda: C_c(X) \rightarrow \mathbb{C}$ (or $\mathbb{R}$) is a continuous linear functional, then there exists a unique regular complex Borel measure $\mu$ (or real Borel measure, depending on the field) on $X$ such that

$$\Lambda(f) = \int_X f d\mu$$

for all $f \in C_c(X)$.

Interpretations

• The Hilbert spaces version implies that every continuous linear functional can be represented using the inner product with a specific vector in that space.

• The measures version indicates that continuous linear functionals on $C_c(X)$ can be represented as integrals against a unique measure.

Examples and Applications

Legendre Polynomials

Let's consider the space $L^2([-1, 1])$ of square-integrable functions on the interval $[-1, 1]$ with the inner product defined as

$$ \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) , dx $$

We'll look at the Legendre polynomials $P_n(x)$, which are orthogonal with respect to this inner product:

$$ \langle P_m, P_n \rangle = \int_{-1}^{1} P_m(x) P_n(x) , dx = \delta_{mn} $$

where $\delta_{mn}$ is the Kronecker delta. Now, let's say we have a continuous linear functional $f: L^2([-1, 1]) \to \mathbb{R}$. By the Riesz Representation Theorem, there exists a unique $g \in L^2([-1, 1])$ such that

$$ f(h) = \langle h, g \rangle $$

for all $h \in L^2([-1, 1])$. We can expand $g$ in terms of the Legendre polynomials:

$$ g(x) = \sum_{n=0}^{\infty} a_n P_n(x) $$

Then,

$$ f(h) = \langle h, g \rangle = \langle h, \sum_{n=0}^{\infty} a_n P_n \rangle = \sum_{n=0}^{\infty} a_n \langle h, P_n \rangle $$

Here, $a_n$ can be found as:

$$ a_n = \langle g, P_n \rangle $$