Lp Spaces - Applications

TheoremRiesz Representation Theorem for L2

Let $(X, \mathcal{F}, \mu)$ be a $\sigma$ -finite measure space. For every bounded linear functional $\varphi: L^2(\mu) \to \mathbb{R}$ (or $\mathbb{C}$ ), there exists a unique $g \in L^2(\mu)$ such that: $\varphi(f) = \int_X fg \, d\mu \text{ for all } f \in L^2$

Moreover, $\|\varphi\| = \|g\|_2$ , where $\|\varphi\| = \sup_{\|f\|_2 \leq 1} |\varphi(f)|$ is the operator norm.

This establishes that the dual space of $L^2$ is (isometrically isomorphic to) $L^2$ itself.

This theorem shows that $L^2$ is self-dual: every continuous linear functional can be represented as an inner product with an element of the space. This property, combined with completeness, makes $L^2$ a Hilbert space.

ExampleApplication to Quantum Mechanics

In quantum mechanics, states are represented by unit vectors in a Hilbert space $\mathcal{H} = L^2(\mathbb{R}^3)$ , and observables are represented by self-adjoint operators.

The Riesz Representation Theorem ensures that for any measurement (linear functional) $\langle \psi | \cdot \rangle$ on $\mathcal{H}$ , there is a unique state $\psi \in \mathcal{H}$ representing it: $\langle \psi | \phi \rangle = \int_{\mathbb{R}^3} \overline{\psi(x)} \phi(x) \, dx$

This mathematical framework underlies the bra-ket notation of Dirac.

Remark

Extensions and generalizations:

General $L^p$ duality: For $1 < p < \infty$ , the dual of $L^p$ is $L^q$ where $1/p + 1/q = 1$ . Every bounded linear functional on $L^p$ is represented by integration against an $L^q$ function.
$L^1$ dual: The dual of $L^1$ is $L^\infty$ , but the converse is false: the dual of $L^\infty$ is properly larger than $L^1$ (it includes finitely additive measures).
Reflexivity: $L^p$ is reflexive for $1 < p < \infty$ , meaning the dual of the dual equals $L^p$ . However, $L^1$ and $L^\infty$ are not reflexive.
Weak and weak topologies*: The Riesz Representation Theorem enables the study of weak convergence: $f_n \rightharpoonup f$ weakly in $L^2$ if $\langle f_n, g \rangle \to \langle f, g \rangle$ for all $g \in L^2$ .

The Riesz Representation Theorem is central to functional analysis, providing the foundation for studying differential equations, optimization problems, and spectral theory in infinite-dimensional spaces.