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

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A sublinear functional on a real vector space

Definition

Let $X$ be a real vector space (Vector space over a field). A function $p : X \to R$ is a sublinear functional if

$p (x + y) \leq p (x) + p (y) for all x, y \in X,$

and

$p (t x) = t p (x) for all x \in X and all t \geq 0.$

Remarks

Taking $t = 0$ shows $p (0) = 0$ .
No condition is imposed for negative scalars beyond what follows from the two displayed axioms.
A norm on a real vector space is sublinear, but a sublinear functional need not be symmetric: in general one can have $p (- x) \neq p (x)$ .

Depends on

Vector space over a field

Used by

Dependency tree · two levels

8 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, Theorem 26.1 (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, Section 4.2 (standard reference, not scraped)