Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Closed graph theorem

Statement

Assume DC. For Banach spaces X,Y and an everywhere-defined linear T:X→Y, T is bounded if and only if its graph (The graph of a linear operator with a linear domain) is closed.

Facts & Assumptions

Given: DC, Banach spaces X,Y, and an everywhere-defined linear map T.

Proof

technique · direct
1.1

If T is bounded, xn→x and Txn→y, continuity gives y=Tx, so the graph is closed.

given
1.2

Conversely, a closed graph is Banach by A closed subspace of a Banach space is Banach, since X×Y is Banach by Finite products of Banach spaces are Banach.

given
2.1

The first projection G(T)→X is a bounded linear bijection; its inverse is bounded by Bounded inverse theorem. Composing that inverse with the second projection makes T bounded.

step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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