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A note on the intersection of Banach subspaces. (English) Zbl 1071.46017

In this article, the authors prove that a Banach space \(X\) is reflexive if and only if, for every \(x \in X\) and every decreasing sequence \((X_n)\) of closed subspaces of \(X\), \(d(x,\bigcap_{n=1}^\infty X_n) = \lim_{n\to\infty} d(x,X_n)\) where \(d(x,C)\) denotes the distance from \(x\) to \(C\).

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
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[1] Fabian, M., Habala, P., Hájek, P., Pelant, P., Montesinos, V. and Zizler, V., Infinite Dimensional Topology and Functional Analysis, Canadian Mathematical Society, CMS Books in Mathematics, Springer Verlag, 2001. · Zbl 0981.46001
[2] Floret, K., Weakly compact sets, LNM, 801, Springer-Verlag, 1980. · Zbl 0437.46006
[3] James, R. C., Reflexivity and the supremum of linear functionals, Ann. of Math., 66 (1957), 159-169. · Zbl 0079.12704 · doi:10.2307/1970122
[4] , Characterizations of reflexivity, Studia Math., 23 (1964), 205-216. · Zbl 0113.09303
[5] Handbook of the Geometry of Banach Spaces, Vol. I and II, W. B. Johnson and J. Lindenstrauss (Ed.), Elsevier, 2001.
[6] Klee, V., Some characterizations of reflexivity, Rev. Ci. Lima, 52 (1950), 15-23. · Zbl 0040.35403
[7] Köthe, G., Topological Vector Spaces I, GMW, 107, Springer-Verlag, 1966.
[8] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I, EMG, 92, Springer- Verlag 1977. · Zbl 0362.46013
[9] Milman, D. P. and Milman, V. D., Some geometric properties of non-reflexive spaces, Soviet Math. Dokl., 4 (1963), 1250-1252.
[10] Nikolsky, V. N., Some properties of reflexive spaces, Ucz. Zap. Kalininsk. gos. pied. in-ta., 29 (1963), 121-125 (in russian).
[11] Rudin, W., Fourier Analysis on Groups, Interscience Publishers, New York, 1967. · Zbl 0698.43001
[12] Singer, I., Best Approximation in Normed Linear Spaces by elements of Linear Subspaces. Springer-Verlag, New York, 1970. · Zbl 0197.38601
[13] Tyuremskih, I. S., Some properties of the Tchebychev subspaces of a Banach space, Ucz. Zap. Kalininsk. gos. pied. in-ta., 39 (1964), 53-64 (in russian).
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