LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passverified 2026-09-10 (Codex)

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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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A complete normed subspace is closed under countable choice

Statement

Assume countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $V$ be a normed space and let $W \subseteq V$ be a normed subspace. If $W$ is complete for its restricted norm, then $W$ is closed in $V$ .

Facts & Assumptions

Given: A normed space $V$ and a normed subspace $W \subseteq V$ .

[A1]

Countable choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Completeness for the norm metric is the Banach-space condition (Banach space).

[L1]

The normed-subspace metric is the ambient metric restricted to $W \times W$ , and the inclusion is an isometric embedding (Normed subspace).

[L2]

Under countable choice, a complete subspace of any metric space is closed in the ambient space (Closed subspaces of complete metric spaces are complete; the converse under countable choice, claim 1). Its proof uses choice only to select, for each fixed adherent point, a sequence of points in the subspace from shrinking nonempty balls. It does not require completeness of the ambient space.

Proof

technique · direct

1.1L1

By [L1], the metric on $W$ induced by the restricted norm is exactly the ambient norm metric restricted to $W$ .

2.1step 1.1F1given

The hypothesis that $W$ is complete for its restricted norm therefore says that $W$ is complete as a metric subspace of $V$ .

3.1step 2.1A1L2∎

Apply [L2] under [A1]. For each fixed point in the closure, [A1] supplies the approximating sequence; completeness and uniqueness of its ambient limit then put that point in $W$ . Thus $W$ is closed in $V$ . No simultaneous selection of sequences for all closure points is required.

Depends on

Used by

Dependency tree · two levels

21 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

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Kyriakos Keremedis and Eliza Wajch, On densely complete metric spaces and extensions of uniformly continuous functions in ZF (standard reference, not scraped)