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.


 * $(1): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
 * $(2): \quad \norm A = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne \mathbf 0_H}$
 * $(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
 * $(4): \quad \norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$