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:


 * $(1):\qquad L$ is continuous
 * $(2):\qquad L$ is continuous at $\mathbf 0$
 * $(3):\qquad L$ is continuous at some point
 * $(4):\qquad \exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$