Abstract
We introduce the categories Vec p of p-normed vector spaces, Ban p of p -Banach spaces, AC p of p -absolutely and TC p of p -totally convex spaces (0 < p ≤ 1). It will be shown that TC p (AC p ) is the Eilenberg–Moore category of Ban p (Vec p ). Then congruence relations on TC p (AC p )-spaces are studied. There are many differences between TC p (AC p )-spaces and totally (absolutely) convex spaces (i.e. p = 1) (Pumplün and Röhrl, 1984, 1985), which will become apparent in Section 4.
Similar content being viewed by others
References
Adámek J., Herrlich H., and Strecker G. E.: Abstract and Concrete Categories, Wiley Interscience, New York, 1990.
Börger R. and Kemper R.: Cogenerators for convex spaces, Appl. Categ. Structures 2 (1994), 1–11.
Börger R. and Kemper R.: Normed totally convex spaces, Comm. Algebra 21(9) (1993), 3243–3258.
Jarchow H.: Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.
Köthe G.: Topological Vector Spaces I, Springer-Verlag, Berlin, 1969.
Linton F. E. J.: Autonomous equational categories, J. Math. Mech. 15(4) (1966), 637–642.
Manes E. G.: Algebraic Theories, Springer, New York, Heidelberg, Berlin, 1976.
Pumplün D. and Röhrl H.: Banach spaces and totally convex spaces I, Comm. Algebra 12 (1984), 953–1019.
Pumplün D. and Röhrl H.: Banach spaces and totally convex spaces II, Comm. Algebra 13 (1985), 1047–1113.
Pumplün D.: Eilenberg-Moore-algebras revisited, Seminarberichte Fachbereich Mathematik, Fernuniversität, Hagen, 29 (1988), 97–144.
Wickenhäuser A.: Positively convex spaces II, Seminarberichte Fachbereich Mathematik, Fernuniversität, Hagen, 32 (1988), 53–104.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kemper, R. p-Banach Spaces and p-Totally Convex Spaces. Applied Categorical Structures 7, 279–295 (1999). https://doi.org/10.1023/A:1008653724200
Issue Date:
DOI: https://doi.org/10.1023/A:1008653724200