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:

\(\ds \norm A\)

\(\le\)

\(\ds c\)

Definition of Infimum

\(\ds \)

\(<\)

\(\ds \infty\)

As $c \in \R_{> 0}$

The result follows.

$\blacksquare$