Continuity of Linear Transformations

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

Then the following four statements are equivalent:


 * $(1): \quad A$ is continuous
 * $(2): \quad A$ is continuous at $\mathbf{0}_H$
 * $(3): \quad A$ is continuous at some point
 * $(4): \quad \exists c > 0: \forall h \in H: \left\Vert{Ah}\right\Vert_K \le c \left\Vert{h}\right\Vert_H$