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: