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.

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.

Banach space

Definition

Let $V$ be a normed space over $K \in {R, C}$ , in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms for $R$ and Complex norm and complex normed space for $C$ , with induced metric $d (x, y) = ∥ x - y ∥$ (in the complex case supplied by Realification preserves a complex norm and its metric). Then $V$ is a Banach space when this metric space is complete in the sense of Complete metric space: every Cauchy sequence converges in the space.

Equivalently: every Cauchy sequence in $V$ for the norm metric converges to a point of $V$ .

Remarks

The metric is part of the data. Completeness is always completeness for the metric induced by the named norm.
The scalar field is part of the normed-space data; completeness is the same metric condition in either case.

Depends on

Dependency tree · 2 levels

Sources