Norm on Bounded Linear Transformation is Finite

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.

Let $A: X \to Y$ be a bounded linear transformation.

Let $\norm A$ denote the norm of $A$ defined by:
 * $\norm A = \inf \set {c > 0: \forall h \in X : \norm {A x}_Y \le c \norm x_X}$

Then:
 * $\norm A < \infty$

Proof
By definition of a bounded linear transformation:
 * $\exists c \in \R_{> 0}: \forall x \in X : \norm{A x}_Y \le c \norm x_X$

Hence:
 * $\set {\lambda > 0: \forall x \in X : \norm {A x}_Y \le \lambda \norm x_X} \ne \O$

By definition:
 * $\set {\lambda > 0: \forall x \in X : \norm {A x}_Y \le \lambda \norm x_X}$ is bounded below.

From the Greatest Lower Bound Property:
 * $\norm A = \inf \set {\lambda > 0: \forall x \in X : \norm {A x}_Y \le \lambda \norm x_X}$ exists.

We have:

The result follows.