In linear algebra, a sublinear function (or functional as is more often used in functional analysis), 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 <math>X</math> be a vector space over a field <math>\mathbb{K},</math> where <math>\mathbb{K}</math> is either the real numbers <math>\Reals</math> or complex numbers <math>\C.</math>

A function <math>p \colon X \to \mathbb{R}</math> is called a ' if it has these two properties:

<ol>

<li>Positive homogeneity, that is <math>p(r x) = r p(x)</math>, for all <math>r \geq 0</math> and <math>x \in X</math>.

</li>

<li>Subadditivity, that is <math>p(x + y) \leq p(x) + p(y)</math> for <math>x, y \in X.</math>

</li>

</ol>

A function <math>p : X \to \Reals</math> is called or if <math>p(x) \geq 0</math> for all <math>x \in X,</math> although some authors define to instead mean that <math>p(x) \neq 0</math> whenever <math>x \neq 0;</math> these definitions are not equivalent.

It is a if <math>p(-x) = p(x)</math> for all <math>x \in X.</math>

Every subadditive symmetric function is necessarily nonnegative.

That is, <math>f</math> 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 <math>f(n) \in o(n)</math> can be upper-bounded by a concave function of sublinear growth.

See also

Notes

Proofs

References

Bibliography

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