Equivalence of Definitions of Norm of Linear Transformation/Lemma

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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

$\blacksquare$