DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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.

Weak, strict, and strong separation

Definition

For nonempty $A, B \subseteq X$ , a nonzero $f \in X^{*}$ weakly separates $A$ and $B$ if $\sup_{a \in A} Re f (a) \leq \inf_{b \in B} Re f (b)$ . It strictly separates them if $Re f (a) < Re f (b)$ for every $a \in A, b \in B$ , and strongly separates them if there are $α < β$ with $Re f (a) \leq α < β \leq Re f (b)$ for all $a, b$ . Over $R$ , $Re f = f$ ; over $C$ these real parts are essential, since complex values are not ordered.

Depends on

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

Used by

A closed convex set is an intersection of closed half-spaces Corollary
Two closed convex sets can have no strong separator Counterexample
Mazur theorem: weak and norm closure agree for convex sets Theorem
Separate a point from an open convex set Theorem
Separation of disjoint convex sets when one is open Theorem
Strong separation of a closed and a compact convex set Theorem

Dependency tree · two levels

5 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 Buehler and Dietmar Salamon, Functional Analysis, §2.3.3 (standard reference, not scraped)