CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

c0 is not isometrically a dual space

Statement

Assume the Axiom of Choice. Over $R$ or $C$ , $c_{0}$ is not linearly isometric onto the dual of any normed space.

Facts & Assumptions

Given: AC and the real or complex space $c_{0}$ .

[F1]

The closed unit ball of $c_{0}$ has no extreme points (The c0 unit ball has no extreme points).

[F2]

Under AC, the closed dual unit ball of every nonzero normed space has an extreme point (Dual unit ball has extreme points).

Proof

technique · contradiction

1.1givenassume-contra

Assume for contradiction that $T : c_{0} \to Y^{*}$ is a surjective linear isometry. The space $Y$ cannot be zero, because then $Y^{*} = {0}$ whereas $c_{0}$ contains a nonzero coordinate vector.

2.1givenstep 1.1

The isometry maps $B_{c_{0}}$ bijectively onto $B_{Y^{*}}$ . It preserves extreme points in both directions: applying $T$ or $T^{- 1}$ to a strict convex representation preserves its coefficient and turns trivial endpoint equality into trivial endpoint equality.

3.1F1F2step 1.1step 2.1discharge-contradiction∎

By [F2], $B_{Y^{*}}$ has an extreme point, whose inverse image under $T$ is extreme in $B_{c_{0}}$ by step 2.1. This contradicts [F1], so the assumed surjective linear isometry does not exist.

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

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)