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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Coordinate vectors do not converge weakly to zero in ell one

Statement refuted

The coordinate vectors of real or complex $ℓ^{1}$ converge weakly to zero. In fact they have no weak limit.

Facts & Assumptions

[F1]

The $ℓ^{1}$ norm is the sum of absolute values ( $ℓ^{p}$ is the $L^{p}$ space of counting measure), applied to real moduli in the complex case.

[F2]

Weak convergence requires convergence of every bounded scalar-linear functional (Weak convergence of nets and sequences).

Counterexample

Given: $e_{n}$ with value one at coordinate $n$ and zero elsewhere in $ℓ^{1}$ .

1.1

Define $F (x) = \sum_{k = 0}^{\infty} x_{k}$ . Absolute convergence gives a scalar sum, linearity by limits of finite sums, and $∣ F (x) ∣ \leq \sum_{k} ∣ x_{k} ∣ = ∥ x ∥_{1}$ . Thus $F \in (ℓ^{1})^{*}$ , but $F (e_{n}) = 1$ for every $n$ , whereas $F (0) = 0$ . Therefore $e_{n}$ cannot converge weakly to zero.

givenF1F2

2.1

More generally coordinate evaluation $P_{k} (x) = x_{k}$ is bounded since $∣ x_{k} ∣ \leq ∥ x ∥_{1}$ . If $e_{n} ⇀ x$ , then $x_{k} = \lim_{n} P_{k} (e_{n}) = 0$ for each fixed $k$ , so $x = 0$ , already excluded by step 1.1. Thus there is no weak limit. $□$

step 1.1F1F2

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)