RemarkRemark: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The continuous dual, its completeness, and evaluation

Remark

Let $K = R$ or $C$ . For a normed $X$ , its continuous dual is $X^{*} = B (X, K)$ with the operator norm, as in The dual space X^* of a normed space and its dual norm. Since the scalar field is complete, If (Y) is Banach then (\mathcal B(X,Y)) is Banach makes $X^{*}$ Banach even when $X$ is incomplete. The pairing $⟨ f, x ⟩ = f (x)$ is bilinear; it is not an inner product on $X$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, §§1.3.1–1.3.2, Theorem 1.31 and (1.22), pp.31–32 (standard reference, not scraped)