Riesz Representation Theorem (Hilbert Spaces)/Corollary

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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