TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.
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The canonical image of James space has codimension one

Statement

Assume Countable Choice. The canonical image of $J$ is a closed subspace of codimension one in $J^{* *}$ . In particular, $J$ is not reflexive.

Facts & Assumptions

[A1]

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

[L1]

Under Countable Choice, $J^{* *}$ is isometrically $J \oplus R 1$ , and the canonical image is the zero-constant summand (Dual and bidual models for James space).

[L2]

Reflexivity means surjectivity of the canonical map into the bidual (Reflexivity is surjectivity of the canonical map).

Proof

technique · direct

Given: The objects and hypotheses in the Statement.

1.1A1L1

By [A1] and [L1], the quotient of $J^{* *}$ by its canonical $J$ summand is [given, A1, L1] identified by $x + λ 1 \mapsto λ$ with $R$ . The scalar $λ = \lim_{n} z_{n}$ satisfies $∣ λ ∣ \leq ∥ z ∥_{J^{* *}}$ because singleton endpoint variations are $∣ z_{n} ∣$ , so the zero-constant summand is closed.

2.1L1L2step 1.1∎

The constant sequence $1$ has bidual norm one and is not in the [given, L2, L1, step 1.1] canonical image, since elements of $J \subset c_{0}$ tend to zero. Thus the quotient is nonzero and exactly one-dimensional. The canonical map is not surjective, so [L2] says $J$ is not reflexive.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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 Bühler and Dietmar Salamon, Functional Analysis (standard reference, not scraped)