Locally convex approach spaces

M. Sioen, S. Verwulgen

Abstract

We continue the investigation of suitable structures for quantified functional analysis, by looking at the notion of local convexity in the setting of approach vector spaces as introduced in [6]. We prove that the locally convex objects are exactly the ones generated (in the usual approach sense) by collections of seminorms. Furthermore, we construct a quantified version of the projective tensor product and show that the locally convex objects admitting a decent exponential law with respect to it are precisely the seminormed spaces.


Keywords

Approach vector space; Topological vector space; Locally convex space; Locally convex approach space; Minkowski functional; Minkowski system; Projective tensor product

Full Text:

PDF

References

Adámek J., Herrlich H. and Strecker G., Abstract and concrete categories, J. Wiley and Sons, 1990.

Borceux F. Handbook of Categorical Algebra, Vol. I,II,III, Encyclopedia of Math. And its Appl., Cambridge University Press, 1996.

Bourbaki N. Eléments de mathématique, livre V, Espaces Vectoriels Topologiques, Hermann, Paris 1964.

Grothendieck A. Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math Soc. 16,1955.

Lowen R. and Sioen M. Approximations in Functional Analysis, Result. Math. 37 (2000) 345-372. http://dx.doi.org/10.1007/BF03322005

Lowen R. and Verwulgen S. Approach Vector Spaces, submitted.

Lowen R. and Windels B. Approach groups, Rocky Mountain J. of Math 30(3) (2000) 1057-1074. http://dx.doi.org/10.1216/rmjm/1021477259

Lowen R. Approach spaces A common Supercategory of TOP and Met, Math. Nachr. 141 (1989) 183-226. http://dx.doi.org/10.1002/mana.19891410120

Lowen R. Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997.

Mac Lane S. Categories for the working mathematician, Springer 1997.

Pumplün D. and Röhrl H. Banach Spaces and Totally Convex Spaces I, Communications in algebra, 12(8) (1984) 953-1019. http://dx.doi.org/10.1080/00927878408823035

Pumplün D. and Röhrl H. Banach Spaces and Totally Convex Spaces II, Communications in algebra 13(5) (1985) 1047-1113. http://dx.doi.org/10.1080/00927878508823205

Pumplun D. Eilenberg-Moore algebras revisited, Seminarberichte, FB Mathematik und Informatic, Fernuniversität 29 (1988) 57-144.

Rudin W. Functional analysis, Intern. Series in Pure and Appl. Math., McGraw-hill, 1991.

Ryan R.A. Introduction to Tensor Product of Banach spaces, Springer Monographs in Math., Springer Verlag (London), 2002. http://dx.doi.org/10.1007/978-1-4471-3903-4

Schaefer H.H. Topological vector spaces, Graduate Texts in Mathematics, Springer, 1999. http://dx.doi.org/10.1007/978-1-4612-1468-7

Sydow W. On Hom-functors and tensor products of topological vector spaces, Lecture notes in Math. 962 (1982) 292-301. http://dx.doi.org/10.1007/BFb0066910

Schatten R. A theory of cross spaces, Annals of math. studies 26, Princeton university press, 1950.

Von Neumann J. On infinite direct products, Compositio Math. 6 (1938) 1-77.

Abstract Views

868
Metrics Loading ...

Metrics powered by PLOS ALM


 

Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. Duality, Vector Spaces and Absolutely Convex Modules
S. Verwulgen
Applied Categorical Structures  vol: 15  issue: 5-6  first page: 647  year: 2007  
doi: 10.1007/s10485-006-9031-x



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt