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.

Absorbing, balanced, and absolutely convex sets

Definition

Let $X$ be a normed space over $K \in {R, C}$ , with the real norm definition A norm on a real vector space, the induced metric, and the dictionary with the metric axioms over $R$ and the complex norm definition Complex norm and complex normed space over $C$ , and let $C \subseteq X$ . The set $C$ is absorbing if, for every $x \in X$ , there is $t > 0$ such that $x \in t C$ . It is balanced if $λ C \subseteq C$ for every $λ \in K$ with $∣ λ ∣ \leq 1$ . It is absolutely convex if it is both convex and balanced. No closedness, openness, or positive definiteness is part of these definitions.

Depends on

Used by

Dependency tree · two levels

16 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

Gerald Teschl, Topics in Real and Functional Analysis, §5.1 (standard reference, not scraped)