DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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 dual space X^* of a normed space and its dual norm

Definition

Let $X$ be a normed space over the scalar field $K$ , where $K = R$ in the literal definition and $K = C$ by the convention of Real and complex scalar conventions for normed spaces. The dual space of $X$ is

$X^{*} : = B (X, K),$

the space of bounded linear functionals on $X$ (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators).

Each $f \in X^{*}$ is in particular a linear functional in the algebraic sense, so $X^{*}$ is a subspace of the algebraic dual from Linear functionals and the algebraic dual $V^{*} = L (V, F)$ .

The dual norm on $X^{*}$ is the operator norm:

$∥ f ∥_{X^{*}} : = ∥ f ∥ = \sup {∣ f (x) ∣ : ∥ x ∥ \leq 1} .$

Remarks

The pairing between $X^{*}$ and $X$ is evaluation: $(f, x) \mapsto f (x)$ .
In this library, $X^{*}$ means the topological dual unless the phrase "algebraic dual" is written explicitly.

Depends on

Used by

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

Daniel Daners, Introduction to Functional Analysis, Definition 25.1 (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, Section 4.2 (standard reference, not scraped)