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Concept
Sublinear function
Summary
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space is a real-valued function with only 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 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 his version of the Hahn-Banach theorem.
There is also a different notion in computer science, described below, that also goes by the name "sublinear function."
Let be a vector space over a field where is either the real numbers or complex numbers
A real-valued function on is called a (or a if ), and also sometimes called a or a , if it has these two properties:
Positive homogeneity/Nonnegative homogeneity: for all real and all
This condition holds if and only if for all positive real and all
Subadditivity/Triangle inequality: for all
This subadditivity condition requires to be real-valued.
A function is called or if for all although some authors define to instead mean that whenever these definitions are not equivalent.
It is a if for all
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 for every unit length scalar (satisfying ) and every
The set of all sublinear functions on denoted by can be partially ordered by declaring if and only if for all
A sublinear function is called if it is a minimal element of under this order.
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