# Riesz representation theorem

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There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.

## Contents

### The Hilbert space representation theorem

This theorem establishes an important connection between a Hilbert space and its (continuous) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic; if the field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field $\mathbb{R}$ or $\mathbb{C}$. If x is an element of H, then the function $\varphi_x$ defined by

where $\langle\cdot,\cdot\rangle$ denotes the inner product of the Hilbert space, is an element of H * . The Riesz representation theorem states that every element of H * can be written uniquely in this form.

Theorem. The mapping

is an isometric (anti-) isomorphism, meaning that:

• Φ is bijective.
• The norms of x and Φ(x) agree: $\Vert x \Vert = \Vert\Phi(x)\Vert$.
• Φ is additive: Φ(x1 + x2) = Φ(x1) + Φ(x2).
• If the base field is $\mathbb{R}$, then Φ(λx) = λΦ(x) for all real numbers λ.
• If the base field is $\mathbb{C}$, then $\Phi(\lambda x) = \bar{\lambda} \Phi(x)$ for all complex numbers λ, where $\bar{\lambda}$ denotes the complex conjugation of λ.