DefinitionDefinition: Literature-sourcedProof: Not applicable

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

Let $X$ be a normed space over the scalar field $K$ , where $K \in {R, C}$ , using A norm on a real vector space, the induced metric, and the dictionary with the metric axioms over $R$ and Complex norm and complex normed space over $C$ . The dual space of $X$ is

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

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:

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

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