# Definition:Bounded Linear Functional

## Definition

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

Then $L$ is said to be a **bounded linear functional** iff

- $\exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$

In view of Continuity of Linear Functionals, a linear functional on a Hilbert space is bounded if and only if it is continuous.

## Also see

- Norm on Bounded Linear Functionals, an important concept for a bounded linear functional
- Bounded Linear Transformation, of which this is a special case

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $I.3.2$