Sublinear function

In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only 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 except 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 his version of the Hahn-Banach theorem.[1]

There is also a different notion in computer science, described below, that also goes by the name "sublinear function."

Definitions

Let X be a vector space over a field 𝔽, where 𝔽 is either the real numbers ℝ or complex numbers ℂ. A real-valued function f : X → ℝ on X is called a sublinear function (or a sublinear functional if 𝔽 = ℝ), and also sometimes called a quasi-seminorm or a Banach functional, if it has all of the following properties:[1]

  1. Positive homogeneity/Nonnegative homogeneity: f (r x) = r f (x) for any real r ≥ 0 and any x ∈ X; and
  2. Subadditivity/Triangle inequality: f (x + y) ≤ f (x) + f (y) for all x, y ∈ X.
    • This subadditivity condition requires f to be real-valued.

A sublinear function f is called positive[2] or nonnegative if f(x) ≥ 0 for all x ∈ X.

The set of all sublinear functions on X, denoted by X#, can be partially ordered by declaring p ≤ q if and only if p(x) ≤ q(x) for all x ∈ X. A sublinear function is called minimal 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.[1]

Examples and sufficient conditions

Every seminorm and norm is a sublinear function a every real linear functional is a sublinear function. The converses are not true in general.

If p and q are sublinear functions on a real vector space X then so is the map x ↦ max { p(x), q(x) }. More generally, if 𝒫 is any non-empty collection of sublinear functionals on a real vector space X and if for all x ∈ X, q(x) ≝ sup { p(x) : p ∈ 𝒫 } < ∞, then q is a sublinear functional on X.[3]

The linear functional x ↦ -x on X = ℝ is a sublinear functional that is not positive and is not a seminorm.[3]

Properties

Every sublinear function is a convex functional.

If p is a real-valued sublinear function on X then:

  • p(0) = 0.[2]
  • 0 ≤ max {p(x), p(-x) } for all x ∈ X.[2]
    • The map defined by q (x) ≝ max {p(x), p(-x)} is a seminorm on X.[2]
    • This implies, in particular, that at least one of p(x) and p(-x) is non-negative.
  • p(x) - p(y) ≤ p(x - y) for all x, y ∈ X.[1]

Associated seminorm

If p is a real-valued sublinear function on X then the map q(x) ≝ max { p(x), p(-x)} defines a seminorm on X called the seminorm associated with p.[2]

Relation to linear functions

If p is a sublinear function on a real vector space X then the following are equivalent:[1]

  1. p is a linear functional;
  2. for every x ∈ X, p(x) + p(-x) ≤ 0;
  3. for every x ∈ 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 ≤ p.[1]

If X is a real vector space, f is a linear functional on X, and p is a sublinear function on X, then f ≤ p on X if and only if f −1(1) ∩ { x ∈ X : p(x) < 1 } = ∅.[1]

Continuity

Theorem[4] — Suppose f : X → ℝ is a subadditive function (i.e. f (x + y) ≤ f (x) + f (y) for all x, y ∈ X). Then f is continuous at the origin if and only if f is uniformly continuous on X. If f satisfies f (0) = 0 then f is continuous if and only if its absolute value |f| : X → [0, ∞) is continuous. If f is non-negative then f is continuous if and only if { x ∈ X : f (x) < 1 } is open in X.

Suppose X is a TVS over the real or complex numbers and p is a sublinear function on X. Then the following are equivalent:[4]

  1. p is continuous;
  2. p is continuous at 0;
  3. p is uniformly continuous on X;

and if p is positive then we may add to this list:

  1. { x ∈ X : p(x) < 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 ≤ p on X implies that f is continuous.[4]

Relation to Minkowski functions

Theorem[4] — If U is a convex open neighborhood of the origin in X then the Minkowski functional of U, pU : X → [0, ∞), is a continuous non-negative sublinear function on X such that U = { x ∈ X : pU (x) < 1  }; if in addition U is balanced then pU is a seminorm on X.

Relation to open convex sets

Theorem[4] — Suppose that X is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of X are exactly those that are of the form z + { x ∈ X : p(x) < 1 } = { x ∈ X : p(x - z) < 1 } for some z ∈ X and some positive continuous sublinear function p on X.

Proof

Let V be an open convex subset of X. If 0 ∈ V then let z := 0 and otherwise let z ∈ V be arbitrary. Let p : X → [0, ∞) be the Minkowski functional of V - z where p is a continuous sublinear function on X since V - z is convex, absorbing, and open (p however is not necessarily a seminorm since V was not assumed to be balanced). From the properties of Minkowski functionals, it is known that V - z = { x ∈ X : p(x) < 1 } from which V = z + { x ∈ X : p(x) < 1 } follows. But

z + { x ∈ X : p(x) < 1 } = { z + x : x ∈ X, p(x) < 1 } = { z + x : x ∈ X, p((z + x) - z) < 1 } = { x ∈ X : p(x - z) < 1  },

as desired. ∎

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 f(n) ∈ o(n) in asymptotic notation (notice the small ). Formally, f(n) ∈ o(n) if and only if, for any given c > 0, there exists an N such that f(n) < cn for n ≥ N.[5] 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) ∈ o(n) can be upper-bounded by a concave function of sublinear growth.[6]

See also

References
  1. Narici & Beckenstein 2011, pp. 177-220.
  2. Narici & Beckenstein 2011, pp. 120-121.
  3. Narici & Beckenstein 2011, pp. 177-221.
  4. Narici & Beckenstein 2011, pp. 192-193.
  5. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001) [1990]. "3.1". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 47–48. ISBN 0-262-03293-7.CS1 maint: multiple names: authors list (link)
  6. Ceccherini-Silberstein, Tullio; Salvatori, Maura; Sava-Huss, Ecaterina (2017-06-29). Groups, graphs, and random walks. Cambridge. Lemma 5.17. ISBN 9781316604403. OCLC 948670194.
    • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
    • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
    • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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