Riesz Isomorphism

Introduction

Let vector space $V$ be a Hilbert space. Let $V^{\ast}$ be the dual space of $V$. Then, according to the Riesz Representation Theorem, any $f \in V^{\ast}$ can be uniquely represented by the following with a unique $\mathbf{v} \in V$.

$$ f = \braket{\cdot, \mathbf{v}} $$

In other words, if you choose $f \in V^{\ast}$, then $\mathbf{v} \in V$ is uniquely determined. Conversely, if $\mathbf{v} \in V$ is chosen, then $\braket{\cdot, \mathbf{v}} = f \in V^{\ast}$ is uniquely determined. Specifically, if $V$ is a Hilbert space, there exists a one-to-one correspondence between $V$ and $V^{\ast}$.

Definition

The following isomorphism between a Hilbert space $V$ and its dual space $V^{\ast}$ is called the Reisz isomorphism.

$$ \begin{align*} \phi_{V} : V &\to V^{\ast} \\ \mathbf{v} &\mapsto \braket{\cdot, \mathbf{v}} \end{align*} $$

Explanation

Note that such an isomorphism exists only when $V$ is a Hilbert space. Since $\phi_{V}$ is an isomorphism, an inverse function exists.

$$ \begin{align*} \phi_{V}^{-1} : V^{\ast} &\to V \\ \braket{\cdot, \mathbf{v}} &\mapsto \mathbf{v} \end{align*} $$