LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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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.

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Norm implies strong implies weak operator convergence

Statement

For nets in $B (X, Y)$ , convergence in operator norm implies strong operator convergence, which implies weak operator convergence. Here $X, Y$ are real or complex normed spaces over the same field. No choice principle is used.

Facts & Assumptions

[F1]

SOT tests $∥ (T_{i} - T) x ∥$ and WOT tests $f ((T_{i} - T) x)$ for each fixed vector and bounded functional (Strong and weak operator topologies).

[F2]

The operator norm satisfies $∥ A x ∥ \leq ∥ A ∥ ∥ x ∥$ , including zero domains (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Proof

Given: a net $T_{i}$ and $T \in B (X, Y)$ .

1.1

If $∥ T_{i} - T ∥ \to 0$ , then for each fixed $x$ , $∥ (T_{i} - T) x ∥ \leq ∥ T_{i} - T ∥ ∥ x ∥ \to 0$ . Explicitly for $ε > 0$ it suffices that $∥ T_{i} - T ∥ < ε / (1 + ∥ x ∥)$ . This is SOT convergence, also at $x = 0$ .

givenF1F2

2.1

If $T_{i} \to T$ in SOT, then for every fixed $x \in X$ and $f \in Y^{*}$ , $∣ f ((T_{i} - T) x) ∣ \leq ∥ f ∥ ∥ (T_{i} - T) x ∥ \to 0$ . The bound $∥ (T_{i} - T) x ∥ < ε / (1 + ∥ f ∥)$ suffices, including $f = 0$ . This is WOT convergence; combined with step 1.1 it proves the hierarchy. $□$

step 1.1F1F2algebra

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

Bühler–Salamon, Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)
Teschl, Topics in Real and Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)