Generating classes of perfect Banach sequence spaces
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- by G. Crofts PDF
- Proc. Amer. Math. Soc. 36 (1972), 137-143 Request permission
Abstract:
A perfect sequence space $\lambda$ is said to be a step if ${l^1} \subset \lambda \subset {l^\infty }$ and $\lambda$ is a Banach space in its strong topology from ${\lambda ^{\text {x}}}$. In this paper a method is given to generate additional steps from a step $\lambda$. Precisely, ${\lambda ^p}$ is a step where ${\lambda ^p} \equiv \{ x = ({x_i})|{x_i} \in C$ and $|x{|^p} = (|{x_i}{|^p}) \in \lambda \}$, for $1 \leqq p < \infty$, with norm ${\left \| x \right \|_{{\lambda ^p}}} = {({\left \| {|x{|^p}} \right \|_\lambda })^{1/p}}$. It is shown that ${\lambda ^p}$, $1 < p < \infty$, is reflexive iff $\lambda$ has a Schauder basis. The space of diagonal maps of ${\lambda ^p}$ into $\lambda$ is characterized, as is the space of diagonal nuclear maps of $\lambda$ into ${\lambda ^p}$ when $\lambda$ has a Schauder basis.References
- G. Crofts, Concerning perfect Fréchet spaces and diagonal transformations, Math. Ann. 182 (1969), 67–76. MR 250015, DOI 10.1007/BF01350165
- Ed Dubinsky, Perfect Fréchet spaces, Math. Ann. 174 (1967), 186–194. MR 220036, DOI 10.1007/BF01360717
- Ed Dubinsky and M. S. Ramanujan, Inclusion theorems for absolutely $\lambda$-summing maps, Math. Ann. 192 (1971), 177–190. MR 293294, DOI 10.1007/BF02052868
- Ed Dubinsky and J. R. Retherford, Schauder bases and Köthe sequence spaces, Trans. Amer. Math. Soc. 130 (1968), 265–280. MR 232184, DOI 10.1090/S0002-9947-1968-0232184-9
- D. J. H. Garling, A class of reflexive symmetric BK-spaces, Canadian J. Math. 21 (1969), 602–608. MR 410331, DOI 10.4153/CJM-1969-068-0
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky-Rogers, Bol. Soc. Mat. São Paulo 8 (1953), 81–110 (1956) (French). MR 94683
- J. R. Holub, Diagonal nuclear maps in sequence spaces, Math. Ann. 191 (1971), 326–332. MR 291864, DOI 10.1007/BF01350335
- 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
- M. S. Ramanujan, Generalised nuclear maps in normed linear spaces, J. Reine Angew. Math. 244 (1970), 190–197. MR 270123, DOI 10.1515/crll.1970.244.190
- William H. Ruckle, Diagonals of operators, Studia Math. 38 (1970), 43–49. (errata insert). MR 275216, DOI 10.4064/sm-38-1-43-49 —, Topologies on sequence spaces, Pacific J. Math. (to appear).
- W. L. C. Sargent, Some sequence spaces related to the $l^{p}$ spaces, J. London Math. Soc. 35 (1960), 161–171. MR 116206, DOI 10.1112/jlms/s1-35.2.161
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- Alfred Tong, Diagonal nuclear operators on $l_{p}$ spaces, Trans. Amer. Math. Soc. 143 (1969), 235–247. MR 251559, DOI 10.1090/S0002-9947-1969-0251559-6
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 36 (1972), 137-143
- MSC: Primary 46A45
- DOI: https://doi.org/10.1090/S0002-9939-1972-0312208-7
- MathSciNet review: 0312208