Equivalence of Definitions of Norm of Linear Transformation

Theorem
Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a bounded linear transformation.

Then the following definitions of the norm of $A$ are equivalent:


 * $(1): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\left\Vert{Ah}\right\Vert_K: \left\Vert{h}\right\Vert_H \le 1}\right\}$
 * $(2): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\dfrac {\left\Vert{Ah}\right\Vert_K} {\left\Vert{h}\right\Vert_H}: h \in H, h \ne \mathbf{0}_H}\right\}$
 * $(3): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\left\Vert{Ah}\right\Vert_K: \left\Vert{h}\right\Vert_H \le 1}\right\}$
 * $(4): \qquad \left\Vert{A}\right\Vert = \inf \left\{{c > 0: \forall h \in H: \left\Vert{Ah}\right\Vert_K \le c \left\Vert{h}\right\Vert_H}\right\}$

Corollary
For all $h \in H$, the following inequality holds:


 * $\left\Vert{Ah}\right\Vert_K \le \left\Vert{L}\right\Vert \left\Vert{h}\right\Vert_H$