DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A bounded bilinear map between normed spaces

Definition

Let $X$ , $Y$ , and $Z$ be normed spaces over the same scalar field $K$ . 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

Used by

For a bilinear map, boundedness is equivalent to joint continuity Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)