Appropriate locally convex domains for differential calculus

Graff, Richard A.; Ruess, Wolfgang M.

doi:10.1090/S0002-9939-1982-0667300-0

Appropriate locally convex domains for differential calculus
HTML articles powered by AMS MathViewer

by Richard A. Graff and Wolfgang M. Ruess PDF

Proc. Amer. Math. Soc. 86 (1982), 331-335 Request permission

Abstract:

We make use of Grothendieck’s notion of quasinormability to produce a comprehensive class of locally convex spaces within which differential calculus may be developed along the same lines as those employed within the class of Banach spaces and which include the previously known examples of such classes. In addition, we show that there exist Fréchet spaces which do not belong to any possible such class.

References

Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168

Elements of non-linear functional analysis

206

Alexandre Grothendieck, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57–123 (French). MR 75542
John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
Wolfgang Ruess, On the locally convex structure of strict topologies, Math. Z. 153 (1977), no. 2, 179–192. MR 454608, DOI 10.1007/BF01179791
W. Ruess, The strict topology and (DF) spaces, Functional analysis: surveys and recent results (Proc. Conf., Paderborn, 1976) North-Holland Math. Studies, Vol. 27; Notas de Mat., No. 63, North-Holland, Amsterdam, 1977, pp. 105–118. MR 0500061
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112