Riesz Representation Theorem (Hilbert Spaces)
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
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 $h \in H$:
| \(\ds L \paren {z L h - h L z}\) | \(=\) | \(\ds L z L h - L h L z\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
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.
$\blacksquare$
Source of Name
This entry was named for Frigyes Riesz.
Sources
As such, the following source works, along with any process flow, will need to be reviewed.
Specifically: with respect to Corollary
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
When you have done this, move the citation of that source work (if it is appropriate that it stay on this page) above this message, into the usual chronological ordering.
When this has been done for all of the sources below, {{SourceReview}} should be removed from the code.
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $I.3.4$