LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Two signs detect an operator increment

Statement

Let $T : X \to Y$ be a bounded linear operator between normed spaces over the same field $K \in {R, C}$ . For all $x, u \in X$ , $\max {∥ T (x + u) ∥, ∥ T (x - u) ∥} \geq ∥ T u ∥ .$ No completeness or choice principle is assumed.

Facts & Assumptions

Given: $T : X \to Y$ as above and $x, u \in X$ .

[F1]

A bounded linear operator is in particular linear, and its domain and codomain have the scalar homogeneity and triangle inequality of normed spaces (A bounded linear operator between normed spaces).

Proof

1.1

By linearity, $T (x + u) - T (x - u) = (T x + T u) - (T x - T u) = 2 T u$ .

F1givenalgebra

2.1

Consequently $2 ∥ T u ∥ = ∥ T (x + u) - T (x - u) ∥ \leq ∥ T (x + u) ∥ + ∥ T (x - u) ∥ \leq 2 \max {∥ T (x + u) ∥, ∥ T (x - u) ∥}$ . Division by the positive real number $2$ proves the claim, including $u = 0$ , $T = 0$ , and zero spaces.

F1step 1.1algebra∎

Remarks

This is the algebraic estimate in Sokal, printed p.2, equation (2), written independently of the surrounding local-ball lemma. Boundedness is part of the operator interface; the calculation needs only linearity.

Depends on

A bounded linear operator between normed spaces

Used by

Sequential uniform boundedness under countable choice Theorem

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

Sokal, A really simple elementary proof of the uniform boundedness theorem (standard reference, not scraped)