Sublinear function


In linear algebra, a sublinear function, 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

Let be a vector space over a field where is either the real numbers or complex numbers
A function is called a ' if it has these two properties:

  1. Positive homogeneity, that is, for all and .
  2. Subadditivity, that is for
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 and
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.
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 on is an example of a sublinear function that is neither positive nor a seminorm; the same is true of this map's negation
More generally, for any real the map
is a sublinear function on and moreover, every sublinear function is of this form; specifically, if and then and
If and are sublinear functions on a real vector space then so is the map More generally, if is any non-empty collection of sublinear functionals on a real vector space and if for all then is a sublinear functional on
A function which is subadditive, convex, and satisfies is also positively homogeneous. If is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming, any two properties among subadditivity, convexity, and positive homogeneity implies the third.

Properties

Every sublinear function is a convex function: For
If is a sublinear function on a vector space then
for every which implies that at least one of and must be nonnegative; that is, for every
Moreover, when is a sublinear function on a real vector space then the map defined by is a seminorm.
Subadditivity of guarantees that for all vectors
so if is also symmetric then the reverse triangle inequality will hold for all vectors
Defining then subadditivity also guarantees that for all the value of on the set is constant and equal to
In particular, if is a vector subspace of then and the assignment which will be denoted by is a well-defined real-valued sublinear function on the quotient space that satisfies If is a seminorm then is just the usual canonical norm on the quotient space
Adding to both sides of the hypothesis and combining that with the conclusion gives
which yields many more inequalities, including, for instance,
in which an expression on one side of a strict inequality can be obtained from the other by replacing the symbol with and moving the closing parenthesis to the right of an adjacent summand.

Associated seminorm

If is a real-valued sublinear function on a real vector space then the map defines a seminorm on the real vector space called the seminorm associated with
A sublinear function on a real or complex vector space is a symmetric function if and only if where as before.
More generally, if is a real-valued sublinear function on a vector space then
will define a seminorm on if this supremum is always a real number.

Relation to linear functionals

If is a sublinear function on a real vector space then the following are equivalent:

  1. is a linear functional.
  2. for every
  3. for every
  4. is a minimal sublinear function.
If is a sublinear function on a real vector space then there exists a linear functional on such that
If is a real vector space, is a linear functional on and is a positive sublinear function on then on if and only if

Dominating a linear functional

A real-valued function defined on a subset of a real or complex vector space is said to be a sublinear function if for every that belongs to the domain of
If is a real linear functional on then is dominated by if and only if
Moreover, if is a seminorm or some other then if and only if

Continuity

Suppose is a topological vector space over the real or complex numbers and is a sublinear function on
Then the following are equivalent:

  1. is continuous;
  2. is continuous at 0;
  3. is uniformly continuous on ;
and if is positive then this list may be extended to include:

  1. is open in
If is a real TVS, is a linear functional on and is a continuous sublinear function on then on implies that is continuous.

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, a function is called sublinear if or in asymptotic notation.
Formally, if and only if, for any given there exists an such that for
That is, 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 can be upper-bounded by a concave function of sublinear growth.