Sublinear function

In linear algebra, a sublinear function (or functional, as is more often used in functional analysis) is a function ${\displaystyle f:V\to \mathbb {F} }$ on a vector space V over ${\displaystyle \mathbb {F} }$ an ordered field (e.g. the real numbers ${\displaystyle \mathbb {R} }$), which satisfies

${\displaystyle f(\gamma x)=\gamma f(x)}$ for any positive ${\displaystyle \gamma \in \mathbb {F} }$ and any ${\displaystyle x\in V}$ (positive homogeneity), and

${\displaystyle f(x+y)\leq f(x)+f(y)}$ for any x, y ∈ V (subadditivity).

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 ${\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} }$ is called sublinear if ${\displaystyle \lim _{n\to \infty }f(n)/n=0}$, or ${\displaystyle f(n)\in o(n)}$ in asymptotic notation (notice the small ${\displaystyle o}$). Formally, ${\displaystyle f(n)\in o(n)}$ if and only if, for any given ${\displaystyle c>0}$, there exists an ${\displaystyle n_{0}}$ such that ${\displaystyle f(n)<c\cdot n}$ for ${\displaystyle n\geq n_{0}.}$[1] That is, ${\displaystyle f}$ 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 ${\displaystyle f(n)\in o(n)}$ can be upper-bounded by a concave function of sublinear growth.[2]