Sublinear function

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.