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There are several wellknown 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 antiisomorphic. 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 or . If x is an element of H, then the function defined by
where 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: .
 Φ is additive: Φ(x_{1} + x_{2}) = Φ(x_{1}) + Φ(x_{2}).
 If the base field is , then Φ(λx) = λΦ(x) for all real numbers λ.
 If the base field is , then for all complex numbers λ, where denotes the complex conjugation of λ.
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