Actual room

This article or the following section is not adequately provided with supporting documents ( e.g. individual evidence ). Information without sufficient evidence could be removed soon. Please help Wikipedia by researching the information and including good evidence.

In mathematics one is real metric space (ger .: proper metric space) a metric space in which all closed , limited subsets compact are.

Actual metric spaces are always complete .

Analogously, a subset of a topological vector space is actually called if all closed, bounded subsets are compact. Here is a subset of a (not necessarily metrizable) topological vector space is limited , if for every area of the zero vector a with are. ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle S\subset W}$ ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle 0\in V}$ ${\displaystyle k>0}$ ${\displaystyle S\subset kU}$

A normalized vector space is (in the sense of both definitions) if and only if it is finite-dimensional .