# Riesz Representation Theorem (Hilbert Spaces)/Corollary

## 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.