Continuity of Linear Functionals

Theorem
Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$.

Then the following four statements are equivalent:


 * $(1): \quad L$ is continuous
 * $(2): \quad L$ is continuous at $\mathbf 0_H$
 * $(3): \quad L$ is continuous at some point
 * $(4): \quad \exists c > 0: \forall h \in H: \size {L h} \le c \norm h$