DefinitionDefinition: Literature-sourcedProof: Not applicable

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A bounded bilinear map between normed spaces

Definition

Let $X$ , $Y$ , and $Z$ be normed spaces over the same scalar field $K \in {R, C}$ , using A norm on a real vector space, the induced metric, and the dictionary with the metric axioms over $R$ and Complex norm and complex normed space over $C$ . A map $B : X \times Y \to Z$ is bilinear when $x \mapsto B (x, y)$ is linear for each fixed $y \in Y$ and $y \mapsto B (x, y)$ is linear for each fixed $x \in X$ .

It is a bounded bilinear map when there is a real constant $C \geq 0$ such that

$∥ B (x, y) ∥ \leq C ∥ x ∥ ∥ y ∥ for all x \in X, y \in Y .$

Remarks

If either variable is $0$ , the displayed estimate forces $B (x, 0) = 0$ and $B (0, y) = 0$ .
The constant $C$ is not unique, exactly as for bounded linear operators.

Depends on

Dependency tree · 2 levels

Sources