Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge 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.
  • 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.

The standard unit vectors are not a Schauder basis of ell-infinity

Statement refuted

The standard unit vectors form a Schauder basis of ℓ∞.

Facts & Assumptions

[L1]

A Schauder basis expansion must converge in norm to every vector (Schauder basis and coordinate functionals).

[L2]

c0 consists of scalar sequences tending to zero and is contained in ℓ∞ (The sequence spaces c_0 and ell-infinity).

Counterexample

technique · counterexample

Given: The objects and hypotheses in the Statement.

1.1

Every finite linear combination of standard unit vectors has finite [given, L2] support. A supremum-norm limit of finite-support sequences lies in c0: for a given tolerance, approximate uniformly by one finite-support sequence and use its finite support to bound the tail.

L2uniform limit
2.1

The constant-one sequence belongs to ℓ∞ but not to c0. [given, L1, L2, step 1.1] Therefore it is not the norm limit of standard-unit-vector partial sums, in violation of [L1]. This explicit witness refutes the statement.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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