Category:Definitions/Sublinear Functionals

This category contains definitions related to Sublinear Functionals.
Related results can be found in Category:Sublinear Functionals.

Let $E$ be a vector space over $\R$.

A real-valued function $p: E \to \R$ is called a sublinear functional if and only if it satisfies:

$(1)$	$:$		Subadditivity:	$\ds \forall x, y \in E:$		$\ds \map p {x + y} $	$\ds \le $	$\ds \map p x + \map p y $
$(2)$	$:$		Positive Homogeneity:	$\ds \forall x \in E, \forall \lambda \in \R_{>0}:$		$\ds \map p {\lambda x} $	$\ds = $	$\ds \lambda \map p x $

Pages in category "Definitions/Sublinear Functionals"

The following 2 pages are in this category, out of 2 total.

\((1)\)	$:$		Subadditivity:	\(\ds \forall x, y \in E:\)		\(\ds \map p {x + y} \)	\(\ds \le \)	\(\ds \map p x + \map p y \)
\((2)\)	$:$		Positive Homogeneity:	\(\ds \forall x \in E, \forall \lambda \in \R_{>0}:\)		\(\ds \map p {\lambda x} \)	\(\ds = \)	\(\ds \lambda \map p x \)