# Riesz representation theorem

The

**Riesz representation theorem** in

functional analysis establishes an important connection between a

Hilbert space and its

dual space: if the ground field is the

real numbers, the two are isometrically isomorphic; if the ground field is the

complex numbers, the two are isometrically anti-isomorphic. The theorem is the justification for the

bra-ket notation popular in the mathematical treatment of

quantum mechanics. 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 functions from *H* into the base field **R** or **C**. If *x* is an element of *H*, then φ_{x} defined by

- φ
_{x}(*y*) = <*x*, *y*> for all *y* in *H*

is an element of

*H* '. The Riesz representation theorem states that

*every* element of

*H* ' can be written in this form, and that furthermore the assignment Φ(

*x*) = φ

_{x}
defines an isometric (anti-) isomorphism

- Φ :
*H* `->` *H* '

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 an 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*||

^{2} ·

*z*. Then Φ(

*x*) = φ.