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CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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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 Banach space with a Schauder basis is separable

Statement

Every real or complex Banach space with a Schauder basis is separable.

Facts & Assumptions

[L1]

Every vector has norm-convergent finite partial sums in the basis (Schauder basis and coordinate functionals).

Proof

technique · direct

Given: The objects and hypotheses in the Statement.

1.1

In the real case let D consist of all finite linear combinations of the [given] basis vectors with rational coefficients. In the complex case use coefficients in Q+iQ. This is a countable union of countable finite products and hence is countable.

definition
2.1

Given x, first use [L1] to choose a basis partial sum within [given, L1, step 1.1] ε/2 of x. Approximate its finitely many scalar coefficients by rational, respectively Gaussian-rational, scalars closely enough that the resulting finite combination changes by less than ε/2. It belongs to D and is within ε of x. Thus D is dense.

L1step 1.1finite approximation∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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