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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, a sublinear function (or functional as is more often used in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
), also called a quasi-seminorm or a Banach functional, on a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
X is a real-valued function with only some of the properties of a
seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
. 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 Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
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 Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, described below, that also goes by the name "sublinear function."


Definitions

Let X be a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over a field \mathbb, where \mathbb is either the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s \Reals or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s \C. A real-valued function p : X \to \mathbb on X is called a ' (or a ' if \mathbb = \Reals), and also sometimes called a ' or a ', if it has these two properties:
  1. '' Positive homogeneity/ Nonnegative homogeneity'': p(r x) = r p(x) for all real r \geq 0 and all x \in X. * This condition holds if and only if p(r x) = r p(x) for all positive real r > 0 and all x \in X.
  2. '' Subadditivity/ Triangle inequality'': p(x + y) \leq p(x) + p(y) for all x, y \in X. * This subadditivity condition requires p to be real-valued.
A function p : X \to \Reals is called or if p(x) \geq 0 for all x \in X, although some authors define to instead mean that p(x) \neq 0 whenever x \neq 0; these definitions are not equivalent. It is a if p(-x) = p(x) for all 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 In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
. 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) \leq p(x) for every unit length scalar u (satisfying , u, = 1) and every x \in X. The set of all sublinear functions on X, denoted by X^, can be partially ordered by declaring p \leq q if and only if p(x) \leq q(x) for all x \in X. A sublinear function is called ' if it is a minimal element of 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 In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
, and real linear functional is a sublinear function. The identity function \Reals \to \Reals on X := \Reals 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 \mapsto -x. More generally, for any real a \leq b, the map \begin S_ :\;&& \Reals &&\;\to \;& \Reals \\ .3ex && x &&\;\mapsto\;& \begin a x & \text x \leq 0 \\ b x & \text x \geq 0 \\ \end \\ \end is a sublinear function on X := \Reals and moreover, every sublinear function p : \Reals \to \Reals is of this form; specifically, if a := - p(-1) and b := p(1) then a \leq b and p = S_. If p and q are sublinear functions on a real vector space X then so is the map x \mapsto \max \. More generally, if \mathcal is any non-empty collection of sublinear functionals on a real vector space X and if for all x \in X, q(x) := \sup \, then q is a sublinear functional on X. A function p : X \to \Reals which is subadditive, convex, and satisfies p(0) \leq 0 is also positively homogeneous (the latter condition p(0) \leq 0 is necessary as the example of p(x):=\sqrt on X:=\mathbb R shows). If p is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming p(0) \leq 0, any two properties among subadditivity, convexity, and positive homogeneity implies the third.


Properties

Every sublinear function is a
convex function In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
: For 0 \leq t \leq 1, \begin p(t x + (1 - t) y) &\leq p(t x) + p((1 - t) y) && \quad\text \\ &= t p(x) + (1 - t) p(y) && \quad\text \\ \end If p : X \to \Reals is a sublinear function on a vector space X then p(0) ~=~ 0 ~\leq~ p(x) + p(-x), for every x \in X, which implies that at least one of p(x) and p(-x) must be nonnegative; that is, for every x \in X, 0 ~\leq~ \max \. Moreover, when p : X \to \Reals is a sublinear function on a real vector space then the map q : X \to \Reals defined by q(x) ~\stackrel~ \max \ is a seminorm. Subadditivity of p : X \to \Reals guarantees that for all vectors x, y \in X, p(x) - p(y) ~\leq~ p(x - y), - p(x) ~\leq~ p(-x), so if p is also symmetric then the reverse triangle inequality will hold for all vectors x, y \in X, , p(x) - p(y), ~\leq~ p(x - y). Defining \ker p ~\stackrel~ p^(0), then subadditivity also guarantees that for all x \in X, the value of p on the set x + (\ker p \cap -\ker p) = \ is constant and equal to p(x). In particular, if \ker p = p^(0) is a vector subspace of X then - \ker p = \ker p and the assignment x + \ker p \mapsto p(x), which will be denoted by \hat, is a well-defined real-valued sublinear function on the quotient space X \,/\, \ker p that satisfies \hat ^(0) = \ker p. If p is a seminorm then \hat is just the usual canonical norm on the quotient space X \,/\, \ker p. Adding b c to both sides of the hypothesis p(x) + a c \,<\, \inf_ p(x + a K) (where p(x + a K) ~\stackrel~ \) and combining that with the conclusion gives p(x) + a c + b c ~<~ \inf_ p(x + a K) + b c ~\leq~ p(x + a \mathbf) + b c ~<~ \inf_ p(x + a \mathbf + b K) which yields many more inequalities, including, for instance, p(x) + a c + b c ~<~ p(x + a \mathbf) + b c ~<~ p(x + a \mathbf + b \mathbf) in which an expression on one side of a strict inequality \,<\, can be obtained from the other by replacing the symbol c with \mathbf (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

If p : X \to \Reals is a real-valued sublinear function on a real vector space X (or if X is complex, then when it is considered as a real vector space) then the map q(x) ~\stackrel~ \max \ defines a
seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on the real vector space X called the seminorm associated with p. A sublinear function p on a real or complex vector space is a symmetric function if and only if p = q where q(x) ~\stackrel~ \max \ as before. More generally, if p : X \to \Reals is a real-valued sublinear function on a (real or complex) vector space X then q(x) ~\stackrel~ \sup_ p(u x) ~=~ \sup \ will define a
seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on X if this supremum is always a real number (that is, never equal to \infty).


Relation to linear functionals

If p is a sublinear function on a real vector space X then the following are equivalent:
  1. p is a linear functional.
  2. for every x \in X, p(x) + p(-x) \leq 0.
  3. for every x \in X, p(x) + p(-x) = 0.
  4. p is a minimal sublinear function.
If p is a sublinear function on a real vector space X then there exists a linear functional f on X such that f \leq p. If X is a real vector space, f is a linear functional on X, and p is a positive sublinear function on X, then f \leq p on X if and only if f^(1) \cap \ = \varnothing.


Dominating a linear functional

A real-valued function f defined on a subset of a real or complex vector space X is said to be a sublinear function p if f(x) \leq p(x) for every x that belongs to the domain of f. If f : X \to \Reals is a real linear functional on X then f is dominated by p (that is, f \leq p) if and only if -p(-x) \leq f(x) \leq p(x) \quad \text x \in X. Moreover, if p is a seminorm or some other (which by definition means that p(-x) = p(x) holds for all x) then f \leq p if and only if , f, \leq p.


Continuity

Suppose X is a topological vector space (TVS) over the real or complex numbers and p is a sublinear function on X. Then the following are equivalent:
  1. p is continuous;
  2. p is continuous at 0;
  3. p is uniformly continuous on X;
and if p is positive then this list may be extended to include:
  1. \ is open in X.
If X is a real TVS, f is a linear functional on X, and p is a continuous sublinear function on X, then f \leq p on X implies that f 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 Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, a function f : \Z^+ \to \Reals is called sublinear if \lim_ \frac = 0, or f(n) \in o(n) in asymptotic notation (notice the small o). Formally, f(n) \in o(n) if and only if, for any given c > 0, there exists an N such that f(n) < c n for n \geq N. That is, f grows slower than any linear function. The two meanings should not be confused: while a Banach functional is convex, almost the opposite is true for functions of sublinear growth: every function f(n) \in o(n) can be upper-bounded by a concave function of sublinear growth.


See also

* * * * * * * *


Notes

Proofs


References


Bibliography

* * * * * * {{Topological vector spaces Articles containing proofs Functional analysis Linear algebra Types of functions