In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm, on a vector space is a real-valued function with some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm except that it is not required to map non-zero vectors to non-zero values. In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem. The notion of a sublinear function was introduced by Stefan Banach when he proved the Hahn-Banach theorem. There is also a different notion in computer science, described below, that also goes by the name "sublinear function."
Definitions Let X {\displaystyle X} be a vector space over a field K , {\displaystyle \mathbb {K} ,} where K {\displaystyle \mathbb {K} } is either the real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} A function p : X → R {\displaystyle p\colon X\to \mathbb {R} } is called a sublinear if it has these two properties:
Positive homogeneity, that is p ( r x ) = r p ( x ) {\displaystyle p(rx)=rp(x)} , for all r ≥ 0 {\displaystyle r\geq 0} and x ∈ X {\displaystyle x\in X} .
Subadditivity, that is p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)} for x , y ∈ X . {\displaystyle x,y\in X.}
A function p : X → R {\displaystyle p:X\to \mathbb {R} } is called positive or nonnegative if p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ X , {\displaystyle x\in X,} although some authors define positive to instead mean that p ( x ) ≠ 0 {\displaystyle p(x)\neq 0} whenever x ≠ 0 ; {\displaystyle x\neq 0;} these definitions are not equivalent. It is a symmetric function if p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} for all x ∈ X . {\displaystyle x\in X.} Every subadditive symmetric function is necessarily nonnegative. A sublinear function on a real vector space is symmetric if and only if it is a seminorm. A sublinear function on a real or complex vector space is a seminorm if and only if it is a balanced function or equivalently, if and only if p ( u x ) ≤ p ( x ) {\displaystyle p(ux)\leq p(x)} for every unit length scalar u {\displaystyle u} and x ∈ X . {\displaystyle x\in X.} The set of all sublinear functions on X , {\displaystyle X,} denoted by X # , {\displaystyle X^{\#},} can be partially ordered by declaring p ≤ q {\displaystyle p\leq q} if and only if p ( x ) ≤ q ( x ) {\displaystyle p(x)\leq q(x)} for all x ∈ X . {\displaystyle x\in X.} A sublinear function is called minimal if it is a minimal element of X # {\displaystyle X^{\#}} under this order. A sublinear function is minimal if and only if it is a real linear functional.
Examples and sufficient conditions Every norm, seminorm, and real linear functional is a sublinear function. The identity function on R {\displaystyle \mathbb {R} } is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation x ↦ − x . {\displaystyle x\mapsto -x.} More generally, for any real a ≤ b , {\displaystyle a\leq b,} the map
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