Hostname: page-component-5c6d5d7d68-thh2z Total loading time: 0 Render date: 2024-08-18T10:32:45.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Köthe function spaces. I

Published online by Cambridge University Press:  24 October 2008

Nguyen Phuong-Các
Nguyen Phuong-Các
Affiliation:
University of Iowa, Iowa City, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

The idea of constructing a space of functions taking values in a locally convex space E from a linear space of scalar valued functions is well known. We can, for example, define a space consisting of all E-valued functions φ(t) such that for all elements e′ of the dual E′ of E. Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function space and the vector space E. Therefore one usually starts from some function space which has interesting properties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 3 , May 1969 , pp. 601 - 611
Copyright
Copyright © Cambridge Philosophical Society 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Banach, S.Théorie des opérations linéaires. Monografje Matematyczne, Tome 1 (Warsaw, 1932).Google Scholar
(2)Bourbaki, N.Eléments de Mathématique. Livre IV. Intégration. Actualités Sci. Ind. 1175 (Paris, 1952 or 1965; Chapters I–IV), 1244 (Paris, 1956; Chapter V), 1281 (Paris, 1958; Chapter VI).Google Scholar
(3)Cooper, J. L. B.Coordinated linear spaces. Proc. London Math. Soc. (3), 3 (1953), 305–327.CrossRefGoogle Scholar
(4)Cooper, J. L. B.On a generalization of the Köthe coordinated spaces. Math. Ann. 162 (1966), 351–363.CrossRefGoogle Scholar
(5)Dieudonné, J.Sur les espaces de Köthe. J. Analyse Math. 1 (1951), 81–115.CrossRefGoogle Scholar
(6)Köthe, G.Topologische lineare Räume 1. Springer-Verlag (Berlin, 1960).CrossRefGoogle Scholar
(7)Pietsch, A.Verallgemeinerte vollkommene Folgenräume. Sch. Forschungsinst. Math., Heft 12 (Berlin, 1962).Google Scholar
(8)Pietsch, A.Verallgemeinerte vollkommene Räume. Studia Math. 1 (1963), 89–91.Google Scholar
(9)Schwartz, L.Espaces de fonctions différentiables à valeurs vectorielles. J. Analyse Math. 4 (1955), 88–148.CrossRefGoogle Scholar
(10)Welland, R.On Köthe spaces. Trans. Amer. Math. Soc. 112 (1964), 267–277.Google Scholar
(11)Nguyen, Phuong Các. On Dieudonné's paper Sur les espaces de Köthe. Proc. Cambridge Philos. Soc. 62 (1966), 29–32.Google Scholar