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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reflexivity of $ L(E,\,F)$

Author: William H. Ruckle
Journal: Proc. Amer. Math. Soc. 34 (1972), 171-174
MSC: Primary 46B10; Secondary 47A99
MathSciNet review: 0291777
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Abstract: Let E and F be two Banach spaces both having the approximation property. The space $ L(E,F)$ is reflexive if and only if (a) both E and F are reflexive, (b) every continuous linear operator from E into F is compact. Thus $ L({l^p},{l^q})$ is reflexive for $ 1 < q < p < \infty $.

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  • [1] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). MR 0075539
  • [2] H. R. Pitt, A note on bilinear forms, J. London Math. Soc. 11 (1936), 174-180.
  • [3] Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from 𝐿^{𝑝}(𝜇) to 𝐿^{𝑟}(𝜈), J. Functional Analysis 4 (1969), 176–214. MR 0250036

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Keywords: Reflexivity, continuous linear operator, compact linear operator, integral operator, nuclear operator
Article copyright: © Copyright 1972 American Mathematical Society