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The Banach-Saks Theorem in C(S)

  • Nicholas R. Farnum (a1)

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A Banach space X has the Banach-Saks property if every sequence (xn ) in X converging weakly to x has a subsequence (xnk ) with (1/pk=1 xnk converging in norm to x. Originally, Banach and Saks [2] proved that the spaces Lp (p > 1) have this property. Kakutani [4] generalized their result by proving this for every uniformly convex Banach space, and in [9] Szlenk proved that the space L 1 also has this property.

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References

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1. Baernstein, A., On reflexivity and summability, Studia Math. 42 (1972), 9194.
2. Banach, et Saks, , Sur la convergence for te dans les champs Lp , Studia Math. 2 (1930), 5157.
3. Dunford, N. and Schwartz, J. T., Linear operators (Interscience, New York-London, 1967).
4. Kakutani, S., Weak convergence in uniformly convex spaces, Tôhoku Math. J. 45 (1938), 188193.
5. Nishiura, T. and D. Waterman, Reflexivity and summability, Studia Math. 23 (1963), 5357.
6. Schreier, J., Ein Gegenbeispiel zur Théorie der schwachen Konvergenz, Studia Math. 2 (1930), 5862.
7. Sierpinski, W., General topology (University of Toronto Press, Toronto, 1961).
8. Singer, I., A remark on reflexivity and summability, Studia Math. 26 (1965), 113114.
9. Szlenk, W., Sur les suites faiblement convergentes dans l'espace L, Studia Math. 25 (1965), 337341.
10. Waterman, D., Reflexivity and summability. II, Studia Math. 82 (1969), 6163.
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The Banach-Saks Theorem in C(S)

  • Nicholas R. Farnum (a1)

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