# Definition:Bounded Linear Functional

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## Definition

Let $\mathbb F$ be a subfield of $\C$.

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\mathbb F$.

Let $f : X \to \mathbb F$ be a linear functional.

We say that $f$ is a **bounded linear functional** if and only if:

- there exists $C > 0$ such that $\cmod {\map f x} \le C \norm x$ for each $x \in X$.

## 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
- Continuity of Linear Functionals shows that a linear functional is
**bounded**if and only if it is continuous.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem: Definition $3.2$

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- 2020: James C. Robinson:
*Introduction to Functional Analysis*... (previous) ... (next) $12.1$: The Dual Space