Sublinear function

A sublinear function, in linear algebra and related areas of mathematics, is a function $f:\; V\; ightarrow\; mathbf\{F\}$ on a vector space "V" over F, an ordered field (e.g. the real numbers $mathbb\{R\}$), which satisfies, for all scalars γ and vectors "x" and "y":$f(gamma\; x)\; =\; vert\; gamma\; vert\; f(x)$ ("positive homogenity"):$f(x\; +\; y)\; le\; f(x)\; +\; f(y)$ ("subadditivity")

In functional analysis the name Banach functional is used for sublinear function, especially when formulating Hahn–Banach theorem.

In computer science, a function $f:\; mathbb\{Z^+\}\; ightarrow\; mathbb\{R\}$ is called sublinear if $f(n)\; in\; o(n)$ in asymptotic notation (Notice the small $,o$). Formally, $f(n)\; in\; o(n)$ if and only if, for any given $,c\; >\; 0$, you can choose an $,n\_0$ such that [cite book | author = Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein | title = Introduction to Algorithms | origyear = 1990 | edition = 2nd edition | year = 2001 | publisher = MIT Press and McGraw-Hill | pages = 47-48 | chapter = 3.1 | id = ISBN 0-262-03293-7]

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This means that for any linear function $f\text{'},$, for sufficiently large input $f,$ grows slower than $f\text{'},$.

Examples

* Every (semi-)norm is a sublinear function. Opposite is not true, because (semi-)norms can have their domain vector space over any field (not necessarily ordered) and must have $mathbb\{R\}$ as their codomain.

Properties

* Every sublinear function is a convex functional.

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.

References