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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 In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
-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 In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
. The notion of a sublinear function was introduced by
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
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 In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; ...
    / 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 In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element ...
    /
    Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
    '': 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 Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
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 In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
or equivalently, if and only if p(u x) \leq p(x) for every
unit length Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
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 In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an ...
of X^ under this order. A sublinear function is minimal if and only if it is a real
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
.


Examples and sufficient conditions

Every
norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...
,
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 Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
\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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
, 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 Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
then the
reverse triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
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 In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...
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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
    .
  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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(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 In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
be, say, an
ordered vector space In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Definition Given a vector space X over the real numbers \Reals and a ...
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 Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Pau ...
(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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
, 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 In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any funct ...
of sublinear growth.


See also

* * * * * * * *


Notes

Proofs


References


Bibliography

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