# 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

$\blacksquare$