CounterexampleConstruction: AI-adaptedVerification: 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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Weak star and weak topologies on a dual can differ

Statement refuted

The weak and weak-star topologies on a normed dual always coincide. They differ on $ℓ^{1} = c_{0}^{*}$ over either the real or complex scalars.

Facts & Assumptions

[F1]

The dual of $c_{0}$ is isometrically $ℓ^{1}$ under the bilinear pairing $a (x) = \sum_{k} a_{k} x_{k}$ (The continuous dual of c0 is ell-one).

[F2]

Weak-star convergence tests vectors of the specified predual (Weak star convergence), whereas weak convergence tests all bounded functionals on the space in question (Weak convergence of nets and sequences).

Counterexample

Given: $e_{n} \in ℓ^{1} = c_{0}^{*}$ , the coordinate unit sequences.

1.1

For every $x \in c_{0}$ , $e_{n} (x) = x_{n} \to 0$ by the definition of $c_{0}$ . Hence $e_{n}$ converges to zero for $σ (ℓ^{1}, c_{0})$ .

givenF1F2

2.1

The scalar-linear map $H (a) = \sum_{k} a_{k}$ on $ℓ^{1}$ is well-defined by absolute convergence and satisfies $∣ H (a) ∣ \leq ∥ a ∥_{1}$ . Thus it is a weakly continuous functional on $ℓ^{1}$ , but $H (e_{n}) = 1$ for all $n$ . The sequence is not weakly null. Equal topologies would have the same convergent sequences and limits, so the weak and the specified weak-star topologies differ. $□$

step 1.1F2algebra

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)