Sublinear function


In linear algebra, a sublinear function is a function on a vector space X over an ordered field ?, that satisfies the following properties:

  1. Positive homogeneity: for any positive and any ; and
  2. Subadditivity: for all.
In functional analysis the name Banach functional is used for sublinear functions, especially when formulating Hahn–Banach theorem.
In contrast, in computer science, a function is called sublinear if, or f ∈ o in asymptotic notation.
Formally, f ∈ o if and only if, for any given c > 0, there exists an N such that f < cn for n ≥ 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 ∈ o can be upper-bounded by a concave function of sublinear growth.

Definitions

A sublinear functional f is called positive if f ≥ 0 for all x ∈ X.
We partially order the set of all sublinear functionals on X, denoted by X#, by declaring p ≤ q if and only if p ≤ q for all x ∈ X.
A sublinear functional is called minimal if it is a minimal element of X# under this order.
It can be shown a sublinear functional is minimal if and only if it is a linear functional.

Examples and sufficient conditions

Properties

If is a real-valued sublinear functional on then:

Associated seminorm

If is a real-valued sublinear functional on then the map defines a seminorm on called the seminorm associated with .

Relation to linear functionals

If is a sublinear functional 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 functional.
If is a sublinear functional 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 sublinear functional on, then on if and only if.

Continuity

Suppose is a TVS over the real or complex numbers and is a sublinear functional on.
Then the following are equivalent:

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

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

Relation to open convex sets

Suppose that is a TVS over the real or complex numbers.
Then the open convex subsets of are exactly those that are of the form for some and some positive continuous sublinear functional on.

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.