LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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The quotient ell-infinity/c_0 has no countable separating family

Statement

Assume ${A C}_{ω}$ . The dual of $ℓ^{\infty} / c_{0}$ has no countable family that separates its points.

Facts & Assumptions

Given: $Q = ℓ^{\infty} / c_{0}$ and an uncountable almost-disjoint family $A$ of infinite subsets of $N$ .

[F1]

$c_{0}$ is closed in $ℓ^{\infty}$ , so the quotient seminorm on $Q$ is a norm (c_0 is a closed subspace of ell-infinity, The quotient seminorm is a norm exactly when the subspace is closed).

[F2]

There is an uncountable almost-disjoint family of infinite subsets of $N$ (An uncountable almost-disjoint family of subsets of the naturals).

[F3]

Assuming ${A C}_{ω}$ , a countable union of countable sets is countable (Countable unions of at most countable sets, assuming ${A C}_{ω}$ ).

Proof

technique · direct

1.1

For $A \in A$ , let $u_{A}$ be its indicator sequence. Then $[u_{A}] \neq 0$ in $Q$ , and for distinct $A_{1}, \dots, A_{m}$ the quotient norm of $\sum c_{j} [u_{A_{j}}]$ is $\max_{j} ∣ c_{j} ∣$ , since the supports are disjoint after deleting finitely many coordinates.

F1F2given

2.1

For $g \in Q^{*}$ and $r > 0$ , the set ${A : ∣ g ([u_{A}]) ∣ \geq r}$ is finite: choose scalar phases on any finite subfamily and apply $∣ g (\sum c_{j} [u_{A_{j}}]) ∣ \leq ∥ g ∥$ . Thus ${A : g ([u_{A}]) \neq 0}$ is countable as a union over positive reciprocal integers.

step 1.1F1algebra

3.1

Given a countable family $(g_{n})$ in $Q^{*}$ , [F3] makes $⋃_{n} {A : g_{n} ([u_{A}]) \neq 0}$ countable. By [F2] choose $A$ outside it; then nonzero $[u_{A}]$ is annihilated by every $g_{n}$ .

step 2.1F2F3choose∎

Depends on

Used by

c₀ is not complemented in ell-infinity Theorem

Dependency tree · two levels

28 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

Piotr Hajlasz, Functional Analysis, proof of Theorem 10.19 (standard reference, not scraped)