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

Uniform boundedness fails on the incomplete space c_00

Statement refuted

Pointwise bounded families on arbitrary normed spaces need not be uniformly operator-norm bounded.

Facts & Assumptions

Given: c00 with the supremum norm and fn(x)=nxn.

Counterexample

technique · direct
1.1

Every fn is bounded and ∥fn∥=n by testing the nth unit vector.

given
1.2

For fixed finitely supported x, fn(x)=0 for all sufficiently large n, so sup⁡n∣fn(x)∣<∞.

given
2.1

The partial sums of (2−k)k≥1 lie in c00 and are Cauchy in the sup norm but converge in its completion to a non-finitely-supported sequence. Thus the domain is incomplete and steps 1.1--1.2 refute the claim.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

5 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