ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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Coordinate evaluations converge weak star to zero in ell one star

Example

For real or complex scalars, the coordinate functionals $E_{n} (x) = x_{n}$ on $ℓ^{1}$ are weak-star null and satisfy $∥ E_{n} ∥ = 1$ . Under $(ℓ^{1})^{*} = ℓ^{\infty}$ , these are the coordinate unit sequences.

Facts & Assumptions

[F1]

The complex identification is the bilinear isometry $b \mapsto (x \mapsto \sum_{k} b_{k} x_{k})$ (The complex continuous dual of ell-one is ell-infinity).

[F2]

The sequence norm is $\sum_{k} ∣ x_{k} ∣$ ( $ℓ^{p}$ is the $L^{p}$ space of counting measure).

[F3]

Weak-star convergence means convergence on each fixed primal vector (Weak star convergence).

Verification

Given: the coordinate functionals on real or complex $ℓ^{1}$ .

1.1

The complex coefficient identification is F1. For real scalars, a bounded sequence $b$ defines $h_{b} (x) = \sum_{k} b_{k} x_{k}$ with $∣ h_{b} (x) ∣ \leq ∥ b ∥_{\infty} ∥ x ∥_{1}$ . Conversely if $h$ is bounded, $b_{k} = h (e_{k})$ satisfies $∣ b_{k} ∣ \leq ∥ h ∥$ ; finite truncations of $x$ converge in norm because their error is the absolute series tail, so $h (x) = \sum_{k} b_{k} x_{k}$ . Testing $e_{k}$ gives $∥ h_{b} ∥ \geq \sup_{k} ∣ b_{k} ∣$ , proving isometry and uniqueness. Thus the identification holds over either field.

givenF1F2

2.1

For every $x \in ℓ^{1}$ , convergence of $\sum_{k} ∣ x_{k} ∣$ forces $x_{n} \to 0$ . Hence $E_{n} (x) \to 0$ for each fixed $x$ , proving weak-star convergence. The inequality $∣ E_{n} (x) ∣ \leq ∥ x ∥_{1}$ and equality $E_{n} (e_{n}) = 1$ give $∥ E_{n} ∥ = 1$ exactly. $□$

step 1.1F2F3

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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)