Norm on Bounded Linear Transformation is Finite
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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.
$\blacksquare$