# Riesz Representation Theorem (Hilbert Spaces)/Corollary

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## Contents

## Corollary to Riesz Representation Theorem (Hilbert Spaces)

Let $H$ be a Hilbert space.

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

Let $h_0 \in H$ be such that:

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

The norm of $L$ satisfies:

- $\norm L = \norm {h_0}$

## Proof

\(\displaystyle \norm L\) | \(=\) | \(\displaystyle \sup_{h \mathop \in H, \, \norm h \mathop = 1} \size {L h}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sup_{h \mathop \in H, \, \norm h \mathop = 1} \size {\innerprod h {h_0} }\) | |||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \sup_{h \mathop \in H, \, \norm h \mathop = 1} \norm h \norm {h_0}\) | Cauchy-Bunyakovsky-Schwarz Inequality | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \norm {h_0}\) |

So:

- $\norm L \le \norm {h_0}$

Also,:

\(\displaystyle \norm {h_0}^2\) | \(=\) | \(\displaystyle \innerprod {h_0} {h_0}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle L h_0\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \norm {h_0}\) | \(=\) | \(\displaystyle \frac {L h_0} {\norm {h_0} }\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle L \frac {h_0} {\norm {h_0} }\) | |||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \sup_{h \mathop \in H, \, \norm h \mathop = 1} \size {L h}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \norm L\) |

So $\norm {h_0} \le \norm L$.

$\blacksquare$

## Source of Name

This entry was named for Frigyes Riesz.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $I.3.4$