Quasi-normalizable space

Quasinormable spaces form a class of locally convex spaces considered in the mathematical subfield of functional analysis . This concept formation, which goes back to A. Grothendieck , allows a characterization of black spaces . The term quasinormabel can also be found in the literature .

definition

A locally convex space is quasinormierbar if for each zero neighborhood another zero environment is, one that at any one limited amount to be found. ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle \lambda >0}$ ${\displaystyle B\subset E}$ ${\displaystyle V\subset B+\lambda U}$

If this condition were even valid for, then a restricted zero neighborhood and thus the space could be normalized . This consideration justifies the name quasinormable . ${\displaystyle \lambda =0}$ ${\displaystyle V}$ ${\displaystyle E}$

Examples

• Normalized spaces can be quasi-normalized, since one can choose a restricted zero neighborhood as in the above definition, for example the open unit sphere. Then applies to everyone , even to . ${\displaystyle V}$ ${\displaystyle V\subset V+\lambda U}$ ${\displaystyle \lambda >0}$ ${\displaystyle \lambda =0}$
• (DF) spaces can be quasi-normalized.
• Schwartz rooms can be quasi-normalized.

One of the characterizations of the Schwartz spaces is precisely that in the above definition one can even choose the bounded set finitely. One can now ask oneself, conversely, which condition a quasi-normalizable space has to fulfill in order to be a Schwartz space. The following sentence applies: ${\displaystyle B}$

• A locally convex space is a Schwartz space if and only if it can be quasi-normalized and every limited set is precompact .

properties

• A countable product of quasi-normalizable spaces can again be quasi-normalized with the product topology .
• Quasi-normable Fréchet rooms are distinguished .

swell

• Reinhold Meise, Dietmar Vogt: Introduction to functional analysis (= Vieweg study 62 advanced course in mathematics ). Vieweg, Braunschweig et al. 1992, ISBN 3-528-07262-8 .
• MP Katz: Every DF-space is quasi-normable , Functional Analysis and Its Applications, Volume 7 (1973), pages 157-158