Abstract
Certain properties E of linear topological or locally convex spaces induce a functor in the corresponding category, which assigns to every space (X,F) an associated topologyF E. The well-known notions of the coarsest barrelled topology stronger than a given locally convex topology or of the strongest locally convex topology weaker than a given linear topology are examples of this concept. In the first two parts of this paper we consider the problem, whether the above functors commute with other processes, such as forming products, linear and locally convex direct sums, inductive limits and completions. With help of two technical lemmas we prove in the third part, that every separated locally convex space is a quotient of a complete locally convex space, in which every bounded set has a finite dimensional linear span. This sharpens results of Y. Kōmura [12], M. Valdivia [18] and W.J. Wilbur [20].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
ADASCH, N.: Über lokaltopologische Vektorräume. Proceedings of the Symposium on Functional Analysis. 51–56, Istambul, Silivri (1973).
BOURBAKI, N.: Espaces Vectoriels Topologiques. Chap. I–V, Paris: Hermann 1966.
BUCHWALTER, H.: Topologies et Compactologies. Publ. Dép. Math. Lyon6-2, 1–74 (1969).
DAZORD, J., JOURLIN, M.: Sur quelques classes d'espaces localement convexes. Publ. Dép. Math. Lyon8-2, 39–69 (1971).
DE WILDE, M.: Vector Topologies and Linear Maps on Products of Topological Vector Spaces. Math. Ann.196, 117–128 (1972).
DE WILDE, M., HOUET, C.: On increasing sequences of absolutely convex sets in locally convex spaces. Math. Ann.192, 257–261 (1971).
DIEROLF, S.: Vererbbarkeitseigenschaften in topologischen Vektorräumen. Dissertation München 1974.
IYAHEN, S.O.: On certain classes of linear topological spaces. Proc. London Math. Soc.18, 285–307 (1968).
IYAHEN, S.O.: On certain classes of linear topological spaces II. J. London Math. Soc3, 609–617 (1971).
KÖHN, J.: Induktive Limiten nicht lokalkonvexer topologischer Vektorräume. Math. Ann.181, 269–278 (1969).
KÖTHE, G.: Topological Vector Spaces I. Berlin-Heidelberg-New York: Springer-Verlag 1969.
KŌMURA, Y.: On linear topological spaces. Kumamoto J. of Science5, 148–157 (1962).
PORTA, H.: Compactly determined locally convex topologies. Math. Ann.196, 91–100 (1972).
ROBERTSON, W.: Completions of topological vector spaces. Proc. London Math. Soc.8, 242–257 (1958).
SWART, J.: Zur Theorie der Schwartz-Räume. Dissertation, ETH Zürich 1973.
VALDIVIA, M.: Absolutely convex sets in barrelled spaces. Ann. Inst. Fourier Grenoble21, 3–13 (1971).
VALDIVIA, M.: Some new results on bornological barrelled spaces. Proceedings of the Symposium on Functional Analysis, 85–90, Istambul, Silivri (1973).
VALDIVIA, M.: Quotients of complete locally convex spaces. Manuscripta math.14, 235–240 (1974).
WAELBROECK, L.: Topological Vector Spaces and Algebras. Lecture Notes in Mathematics, Berlin-Heidelberg-New-York: Springer-Verlag 1971.
WILBUR, W.J.: Reflective and Coreflective Hulls in the Category of Locally Convex Spaces. Gen. Top. and its Appl.4, 235–254 (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dierolf, S. Über assoziierte lineare und lokalkonvexe Topologien. Manuscripta Math 16, 27–46 (1975). https://doi.org/10.1007/BF01169061
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01169061