Definition:Bounded Linear Functional

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