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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.

This article will describe his theorem concerning the dual of a Hilbert space, which is sometimes called the Fréchet-Riesz theorem. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.

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 underlying 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 R or C . If x is an element of H, then the function φ x , for all y in H defined by

φ x ( y ) = y , x ,

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 Φ : HH* defined by Φ ( x ) = φ x is an isometric (anti-) isomorphism, meaning that:

  • Φ is bijective.
  • The norms of x and φ x agree: x = Φ ( x ) .
  • Φ is additive: Φ ( x 1 + x 2 ) = Φ ( x 1 ) + Φ ( x 2 ) .
  • If the base field is R, then Φ ( λ x ) = λ Φ ( x ) for all real numbers λ.
  • If the base field is C, then Φ ( λ x ) = λ ¯ Φ ( x ) for all complex numbers λ, where λ ¯ denotes the complex conjugation of λ.
  • The inverse map of Φ can be described as follows. Given a non-zero element φ of H*, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set x = φ ( z ) ¯ z / z 2 . Then Φ ( x ) = φ .

    Historically, the theorem is often attributed simultaneously to Riesz and Fréchet in 1907 (see references).

    In the mathematical treatment of quantum mechanics, the theorem can be seen as a justification for the popular bra–ket notation. The theorem says that, every bra ψ | has a corresponding ket | ψ , and the latter is unique.

    References

    Riesz representation theorem Wikipedia