Riesz Representation Theorem (Hilbert Spaces)

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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.


Sources