Equivalence of Definitions of Norm of Linear Transformation/Lemma

Theorem

Let $H, K$ be Hilbert spaces.

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

Then:

$\forall \lambda > 0 : \norm{A 0_H}_K = \lambda \norm{0_H}_H$

Proof

Let $\lambda > 0$.

We have:

\(\ds \norm{A 0_H}_K\)

\(=\)

\(\ds \norm{0_K}_K\)

Linear Transformation Maps Zero Vector to Zero Vector

\(\ds \)

\(=\)

\(\ds 0\)

Norm Axiom $\text N 1$: Positive Definiteness

\(\ds \)

\(=\)

\(\ds \lambda \cdot 0\)

\(\ds \)

\(=\)

\(\ds \lambda \norm{0_H}\)

Norm Axiom $\text N 1$: Positive Definiteness

$\blacksquare$