# Riesz Representation Theorem (Hilbert Spaces)

## Theorem

Let $H$ be a Hilbert space.

Let $L$ be a bounded linear functional on $H$.

Then there is a unique $h_0 \in H$ such that:

$\forall h \in H: L h = \innerprod h {h_0}$

## Proof

If $L \equiv 0$ identically, then $L h = 0 = \innerprod h 0$, and the theorem holds.

By Kernel of Bounded Linear Transformation is Closed Linear Subspace, $M := \ker L$ is a closed linear subspace of $H$.

Then we can decompose $H$ as a direct sum:

$H \cong M \oplus M^\perp$

As $L \not \equiv 0$:

$M^\perp \ne \set 0$

Choose a $z \in M^\perp$ with norm $1$.

By linearity of $L$, for any $h \in H$:

 $\ds L \paren {z L h - h L z}$ $=$ $\ds L z L h - L h L z$ $\ds$ $=$ $\ds 0$

So:

$z L h - h L z \in \ker L = M$

Then:

 $\ds L h$ $=$ $\ds L h \innerprod z z$ as $\norm z = 1$ $\ds$ $=$ $\ds \innerprod {z L h} z$ linearity in the first argument $\ds$ $=$ $\ds \innerprod {z L h - h L z + h L z} z$ adding and subtracting $h L z$ in the first argument $\ds$ $=$ $\ds \innerprod {z L h - h L z} z + \innerprod {h L z} z$ linearity in the first argument $\ds$ $=$ $\ds \innerprod {h L z} z$ $z L h - h L z \in M, z \in M^\perp$ $\ds$ $=$ $\ds \innerprod h {z \paren {L z}^*}$ conjugate symmetry

Thus $L h = \innerprod h {h_0}$ for $h_0 = z (Lz)^*$.

To show uniqueness, assume $h_0$ and $h_1$ both satisfy the above equation for all $h \in H$:

 $\ds \innerprod h {h_0}$ $=$ $\ds \innerprod h {h_1}$ $\ds \leadsto \ \$ $\ds \innerprod h {h_0} - \innerprod h {h_1}$ $=$ $\ds 0$ $\ds$ $=$ $\ds \innerprod h {h_0 - h_1}$ additivity in the second argument

The result follows from Setting $h = h_0 - h_1$ and invoking the positive definiteness of the inner product.

$\blacksquare$

## Examples

### $L^2$ Space

Let $\struct{ X, \Sigma, \mu }$ be a measure space.

Let $\map {L^2} \mu$ be the associated $L^2$ space.

Let $F: \map {L^2} \mu \to \GF$ be a bounded linear functional.

Then there exists a unique $f_0 \in \map {L^2} \mu$ such that:

$\ds \forall f \in \map {L^2} \mu: \map F f = \int f \overline{f_0} \rd \mu$

### Space of Square Summable Mappings

Let $\map {\ell^2} \N$ be the space of square summable mappings on $\N$.

Let $N \in \N$.

Let $L_N: \map {\ell^2} \N \to \GF$ be defined by:

$\map {L_N} {\sequence{ a_n } } := a_N$

Let $\delta_N \in \map {\ell^2} \N$ be given by:

$\forall n \in \N: \paren{ \delta_N }_n = \begin{cases} 1 & n = N \\ 0 & n \ne N \end{cases}$

Then for all $a \in \map {\ell^2} \N$:

$\map {L_N} a = \innerprod a {\delta_N}$

## Source of Name

This entry was named for Frigyes Riesz.