# 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:

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

### Corollary

The norm of $L$ satisfies:

$\norm L = \norm {h_0}$

## Proof

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

Otherwise, set:

$M = \map \ker L = \map {L^{-1} } {\set 0}$

Then $M$ is a subspace.

Because $L$ is bounded, it is continuous.

Because $\set 0$ is closed, the continuity of $L$ implies that $M$ is closed.

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 $h \in H$:

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

So:

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

Then:

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

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

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

$\blacksquare$

## Source of Name

This entry was named for Frigyes Riesz.