Norm on Bounded Linear Transformation is Finite

Theorem
Let $H, K$ be Hilbert spaces.

Let $A: H \to K$ be a bounded linear transformation.

Let $\norm A$ denote the norm of $A$ defined by:
 * $\norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$

Then:
 * $\norm A < \infty$

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

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

By definition:
 * $\set {\lambda > 0: \forall h \in H: \norm {A h}_K \le \lambda \norm h_H}$ is bounded below.

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

We have:

The result follows.