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 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

be a vector space is either the real numbers or complex numbers is called a if it has these two properties: Positive homogeneity,. Subadditivity,

is called or although some authors define these definitions are not equivalent. It is a 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 for every unit length

can be partially ordered A sublinear function is called if it is a minimal element 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 the map

which is subadditive, convex, any two properties among subadditivity, convexity, and positive homogeneity implies the third.

Properties

Every sublinear function is a convex function

then is a seminorm.

is also symmetric then the reverse triangle inequality

is a well-defined real-valued sublinear function on the quotient space

(or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).

Associated seminorm

defines a seminorm called the on a real or complex vector space is a symmetric function as before.

will define a seminorm).

Relation to linear functionals

then the following are equivalent: is a linear functional. is a minimal sublinear function.

Dominating a linear functional

is said to be is a real linear functional then is a seminorm or some other

Continuity

is a topological vector space Then the following are equivalent: is continuous; is continuous at 0; ; is positive then this list may be extended to include:

is continuous.

Relation to Minkowski functions and open convex sets

Relation to open convex sets

Operators

The concept can be extended to operators that are homogeneous and subadditive. This requires only that the codomain be, say, an ordered vector space to make sense of the conditions.

Computer science definition

In computer science is called sublinear in asymptotic notation grows slower than any linear function. The two meanings should not be confused: while a Banach functional is convex can be upper-bounded by a concave function of sublinear growth.

See also

Notes

Proofs

References

Bibliography