DefinitionDefinition: Literature-sourcedProof: Not applicable

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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A bounded linear operator between normed spaces

Definition

Let $X$ and $Y$ be normed spaces over the same scalar field $K \in {R, C}$ , read in the real case from A norm on a real vector space, the induced metric, and the dictionary with the metric axioms and in the complex case from Complex norm and complex normed space. A $K$ -linear map $T : X \to Y$ (Linear map between vector spaces over the same field) is a bounded linear operator when there is a real constant $C \geq 0$ such that

$∥ T x ∥ \leq C ∥ x ∥ for all x \in X .$

Any such $C$ is called a bound for $T$ .

Remarks

The zero operator is bounded with bound $0$ .
A bound is not unique: if $C$ works and $C^{'} \geq C$ , then $C^{'}$ works as well.
When $X = {0}$ , every linear map $X \to Y$ is bounded with bound $0$ .

Depends on

Dependency tree · 2 levels

Sources