Equivalence of Definitions of Norm of Linear Transformation/Definition 3 Greater or Equal Definition 4

Theorem
Let $H, K$ be Hilbert spaces.

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

Let:
 * $\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$

and
 * $\lambda_4 = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$

Let:
 * $\lambda_3 \ge \lambda_4$

Lemma
Let $h \in H: h \ne 0_h$.

We have:

and

Hence:
 * $\forall h \in H: h \ne 0_h: \norm{A h}_K \le \lambda_3 \norm h_H$

From Lemma:
 * $\norm {A 0_H}_K = \lambda_3 \norm {0_H}_H$

Hence:
 * $\forall h \in H: \norm{A h}_K \le \lambda_3 \norm h_H$

That is,
 * $\lambda_3 \in \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$

By definition of the infimum:
 * $\lambda_4 \le \lambda_3$