Separated Presheaves of Normed Spaces and \mgreek{W} -valued Norms

Abstract

This paper lays down the axioms of Ω-valued normed spaces and shows their invariance under the so-called tilde-construction. Important examples are given by separated presheaves of normed spaces and by probabilistic normed spaces. Finally, there exists a natural Ω-valued topologization of Ω-valued normed spaces which makes it possible to study continuous linear operators between them.

Keywords:

• -valued vector space
• -valued norm
• -valued topology
• Presheaf

Notes

^†See e.g. Rosenfeld's concept of fuzzy groups (Rosenfeld, 1971).

^‡If (A,E) is complete, then Ω-valued topologies are external descriptions of topologies in the sense of the topos sh(Ω) of sheaves of sets over Ω (Stout, 1975).

Ulrich Höhle is Professor of Mathematics at the Department of Mathematics, Bergische Universität Wuppertal (Germany). His main interests are applications of non-classical logics to general topology and functional analysis. He has written more than 50 papers and is the author of the monography Many Valued Topology and Its Applications. He is a member of the editorial board of Fuzzy Sets and Systems, and is the editor of the following collected volumes: Non-classical Logics and Their Applications To Fuzzy Subsets (Kluwer Academic Publishers, 1995) and Mathematics of Fuzzy Sets (Kluwer Academic Publishers, 1999).