DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-14

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.
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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.

Unconditional and conditional Schauder bases

Definition

A Schauder basis $(e_{n})$ of a Banach space $X$ is unconditional if, for every $x \in X$ , its uniquely determined basis expansion

$\sum_{n = 1}^{\infty} e_{n}^{*} (x) e_{n}$

converges unconditionally. It is conditional if it is not unconditional; equivalently, at least one vector has a basis expansion that is not unconditionally convergent.

The quantifier is over all basis expansions. It does not assert convergence of the formal unweighted series $\sum_{n} e_{n}$ .

Depends on

Used by

The summing basis of c0 is conditional Example

Dependency tree · two levels

4 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

Thomas Schlumprecht, Course Notes in Functional Analysis, Math 655 (standard reference, not scraped)