Minkowski functional

In the mathematical sub-field of functional analysis , the Minkowski functional (after Hermann Minkowski ), often also called gauge functional , is a generalization of the standard concept .

definition

Let it be a topological vector space . If it is an absorbing subset , the function is called ${\displaystyle X}$ ${\displaystyle 0\in U\subseteq X}$

${\displaystyle p_{U}\colon X\to \mathbb {R} _{0}^{+},\quad x\mapsto \inf\{\lambda \mid \lambda \geq 0,x\in \lambda U\}}$

the Minkowski functional or gauge functional . ${\displaystyle U}$

properties

• If the amount to be absorbed is balanced and convex , it is a semi-norm or seminorm . Conversely, the set has the properties mentioned for each seminorm . It follows from this that the locally convex spaces are precisely those spaces whose topology can be defined by a family of seminorms. A locally convex space is Hausdorffian if and only if this family of seminorms is separating. ${\displaystyle U}$ ${\displaystyle p_{U}}$ ${\displaystyle p}$ ${\displaystyle \{x\in X\mid p(x)<1\}}$

• Is a convex, balanced, confined environment of zero, the Minkowski functional is a standard to which the given topology induced . In particular, according to Kolmogoroff's criterion for normalization , a Hausdorff topological vector space can be normalized if and only if there is a bounded convex neighborhood of zero. ${\displaystyle U}$ ${\displaystyle X}$

example

In a Euclidean space (such as the three-dimensional space of everyday perception) one considers the unit sphere as a subset . Then the Minkowski functional is identical to the usual Euclidean norm , because with lies on the edge of the set , i.e. the sphere with radius and center 0. ${\displaystyle U}$ ${\displaystyle \lambda =\|x\|_{2}}$ ${\displaystyle x}$ ${\displaystyle \lambda U}$ ${\displaystyle \lambda }$

Individual evidence

1. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , Chapter I, §6, definition on page 42