ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Ringoid
Semiring
Ring
Left ring action
Module
Linear combination
Linear map
Definition D708
Superlinear map
Formulation 0
Let $R$ be an D5119: Ordered division ring such that
(i) $V$ is a D29: Vector space over $R$
(ii) $W$ is an D1963: Ordered vector space over $R$
(iii) ${\preceq}$ is an D378: Ordering relation on $W$
A D18: Map $f : V \to W$ is superlinear from $V$ to $W$ over $R$ if and only if \begin{equation} \forall \, N \in 1, 2, 3, \ldots : \forall \, x \in V^N : \forall \, r \in R^N : f \left( \sum_{n = 1}^N r_n x_n \right) \succeq \sum_{n = 1}^N r_n f(x_n) \end{equation}