Norm on Bounded Linear Functional is Finite

Theorem
Let $H$ be a Hilbert space.

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

Let $\norm L$ denote the norm on $L$ defined as:
 * $\norm L = \inf \set {c > 0: \forall h \in H: \size {L h} \le c \norm h_H}$

Then:
 * $\norm L < \infty$

Proof
By definition of a bounded linear functional:
 * $\exists c \in \R_{> 0}: \forall h \in H: \size{L h} \le c \norm h_H$

Hence:
 * $\set {\lambda > 0: \forall h \in H: \size {L h} \le \lambda \norm h_H} \ne \O$

By definition: $\set {\lambda > 0: \forall h \in H: \size {L h} \le \lambda \norm h_H}$ is bounded below by $0$.

From the Greatest Lower Bound Property:
 * $\norm L = \inf \set {\lambda > 0: \forall h \in H: \size {L h} \le \lambda \norm h_H}$ exists.

We have:

The result follows.