The Riesz representation theorem

## Linear Methods of Applied Mathematics Evans M. Harrell II and James V. Herod*

#### *(c) Copyright 2000 by Evans M. Harrell II and James V. Herod. All rights reserved.

version of 15 March 2000

## The Riesz representation theorem

Theorem XV.2. (The Riesz representation theorem for Hilbert space.) If {E, < , >} is a Hilbert space, then these are equivalent:

(a) is a continuous, linear function from E to R (or the complex numbers C), and

(b) there is a unique member v of E such that (x) = < x, v > ; for each x in E.

First, let us formalize the

Projection lemma.

Let H be a Hilbert space and M a closed subspace. Let Mperp consist of the vectors which are orthogonal to all the vectors of M. Given any vector x in H, there is a unique vector y in M and a unique z in Mperp such that x = y + z. The function associating y to x is a linear projection operator, as is the function associating z to x.

The Riesz lemma, stated in words, claims that every continuous linear functional comes from an inner product.

Proof of the Riesz lemma:

Consider the null space N = N( ), which is a closed subspace. If N = H, then is just the zero function, and g = 0. This is the trivial case. Otherwise, There must be a nonzero vector in Nperp. In fact, Nperp is one-dimensional, since if there were two linearly independent vectors z1,2 in Nperp, then we can choose numbers a and b, different from 0, such that (a z1-b z2) = a (z1) - b (z2) = 0. But then a z1-b z2 belongs to N and Nperp. This can only happen if z1-b z2=0, which is a contradiction. Conclusion: The entire space Nperp consists of the multiples of a particular non-zero vector z1.

Now scale z1 so that (z1) is a real number and then let

g := (z1) z1/||z1||. Of course, it remains to verify that this g fills the bill. Because of the projection lemma, we may write any x in H as x = a g + (x-a g), where (x-a g) N. Notice also that <x, g> = a <x, g> + 0, because g belongs to Nperp and (x-a g) to N.

We calculate: (x) = a (g) + 0 = < a g + (x-a g), g> = < x, g>

QED