Equivalence of Definitions of Norm of Linear Transformation

Theorem
Let $H, K$ be Hilbert spaces.

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

Proof
Let:
 * $\lambda_1 = \sup \set {\norm {A h}_K: \norm h_H \le 1}$


 * $\lambda_2 = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne 0_H}$


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


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

From Operator Norm is Finite:
 * $\lambda_4 < \infty$

We will show that:
 * $\lambda_4 \ge \lambda_2 \ge \lambda_1 \ge \lambda_3 \ge \lambda_4$

Inequality: $\lambda_1 \ge \lambda_4$
It follows that the definitions are all equivalent.