# Fundamental Property of Norm on Bounded Linear Transformation

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## Theorem

Let $\HH, \KK$ be Hilbert spaces.

Let $A: \HH \to \KK$ be a bounded linear transformation.

Let $\norm A$ denote the norm of $A$ defined by:

- $\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$

Then:

- $\forall h \in \HH: \norm {A h}_\KK \le \norm A \norm h_\HH$

## Proof

From Norm on Bounded Linear Transformation is Finite:

- $\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$ exists

and

- $\norm A < \infty$

Let $x \in \HH \setminus \set{0_\HH}$

Let $\lambda \in \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$.

Then:

\(\ds \norm {A x}_K\) | \(\le\) | \(\ds \lambda \norm x_\HH\) | ||||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \dfrac {\norm {A x}_\KK} {\norm x_\HH}\) | \(\le\) | \(\ds \lambda\) |

As $c$ was arbitrary, then:

- $\forall \lambda \in \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}: \dfrac {\norm {A x}_\KK} {\norm x_\HH} \le \lambda$

By the definition of the infimum:

- $\dfrac {\norm {A x}_\KK} {\norm x_\HH} \le \norm A$

Hence:

- $\norm {A x}_\KK \le \norm A \norm x_\HH$

Since $x$ was arbitrary:

- $\forall h \in \HH \setminus \set {0_\HH}: \norm {A h}_\KK \le \norm A \norm h_\HH$

Lastly, we have:

\(\ds \norm {A 0_\HH}_\KK\) | \(=\) | \(\ds \norm {0_\KK}_\KK\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \norm A \cdot 0\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \norm A \norm {0_\HH}\) |

It follows that:

- $\forall h \in \HH: \norm {A h}_\KK \le \norm A \norm h_\HH$

$\blacksquare$

## Sources

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- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next) $\S \text {II}.1$