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.

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