logo

Semi-linear Function 📂Functions

Semi-linear Function

Definition1

If a function $f : X \to Y$ satisfies the following two conditions, it is called sublinear. For $x,x_{1},x_{2}\in X$ and $a \in \mathbb{R}$,

  • $f(ax) = af(x)$
  • $f(x_{1} + x_{2}) \le f(x_{1}) + f(x_{2})$

Explanation

If the second condition holds as an equality, it is linear, and if it holds as an inequality, it is sublinear.


  1. Robert A. Adams and John J. F. Foutnier, Sobolev Space (2nd Edition, 2003), p54 ↩︎