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

$(4):\quad \exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$

Sources

• John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $I.3.1$