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Boundedly complete $ M$-bases and complemented subspaces in Banach spaces


Authors: William J. Davis and Ivan Singer
Journal: Trans. Amer. Math. Soc. 175 (1973), 187-194
MSC: Primary 46B15
DOI: https://doi.org/10.1090/S0002-9947-1973-0317011-5
MathSciNet review: 0317011
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Abstract: Subsequences of boundedly complete M-bases need not be boundedly complete. An example of a somewhat reflexive space is given whose dual and one of whose factors fail to be somewhat reflexive. A geometric description of boundedly complete M-bases is given which is equivalent to the definitions of V. D. Milman and W. B. Johnson. Finally, certain M-bases for separable spaces give rise to proper complemented subspaces.


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  • [1] L. Alaoglu, Weak topologies of normed linear spaces, Ann. of Math. (2) 41 (1940), 252-267. MR 1, 241. MR 0001455 (1:241e)
  • [2] S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warsaw, 1932.
  • [3] M. M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Heft 21, Reihe: Reelle Funktionen, Springer-Verlag, Berlin, 1958. MR 20 #1187. MR 0094675 (20:1187)
  • [4] D. W. Dean, I. Singer and L. Sternbach, On shrinking basic sequences in Banach spaces, Studia Math, 40 (1971), 23-33. MR 0306876 (46:5998)
  • [5] J. Dixmier, Sur un théorème de Banach, Duke Math J. 15 (1948), 1057-1071. MR 10, 306. MR 0027440 (10:306g)
  • [6] N. Dunford and A. P. Morse, Remarks on the preceding paper of James A. Clarkson, Trans. Amer. Math. Soc. 40 (1936), 415-420. MR 1501881
  • [7] V. F. Gapoškin and M. I. Kadec, Operator bases in Banach space, Mat. Sb. 61 (103) (1963), 2-12. (Russian) MR 27 #1810. MR 0151827 (27:1810)
  • [8] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 0075539 (17:763c)
  • [9] V. I. Gurariĭ and M. I. Kadec, Minimal systems and quasicomplements in Banach space, Dokl. Akad. Nauk SSSR 145 (1962), 255-258 = Soviet Math. Dokl. 3 (1962), 966-968. MR 26 #6728. MR 0149238 (26:6728)
  • [10] R. Herman and R. Whitley, An example concerning reflexivity, Studia Math. 28 (1966/67), 289-294. MR 35 #5900. MR 0215056 (35:5900)
  • [11] R. C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518-527. MR 12, 616. MR 0039915 (12:616b)
  • [12] -, Separable conjugate spaces, Pacific J. Math. 10 (1960), 563-571. MR 22 #8307. MR 0117528 (22:8307)
  • [13] W. B. Johnson, Markushevich bases and duality theory, Trans. Amer. Math. Soc. 149 (1970), 171-177. MR 41 #5927. MR 0261312 (41:5927)
  • [14] -, On the existence of strongly series summable Markushevich bases in Banach spaces, Trans. Amer. Math. Soc. 157 (1971), 481-486. MR 0282201 (43:7914)
  • [15] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. MR 43 #6702. MR 0280983 (43:6702)
  • [16] W. B. Johnson and H. P. Rosenthal, On $ {w^ \ast }$-basic sequences and their applications to the study of Banach spaces, Studia Math. (to appear). MR 0310598 (46:9696)
  • [17] M. I. Kadec and A. Pełczyński, Basic sequences, bi-orthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323. (Russian) MR 31 #6112. MR 0181886 (31:6112)
  • [18] S. Karlin, Bases in Banach spaces, Duke Math. J. 15 (1948), 971-985. MR 10, 548. MR 0029103 (10:548c)
  • [19] J. Lindenstrauss, On James' paper ``Separable conjugate spaces", Israel J. Math. 9 (1971), 279-284. MR 43 #5289. MR 0279567 (43:5289)
  • [20] E. R. Lorch, Bicontinuous linear transformations in certain vector spaces, Bull. Amer. Math. Soc. 45 (1939), 564-569. MR 1, 58. MR 0000346 (1:58c)
  • [21] V. D. Mil'man, The geometric theory of Banach spaces, Uspehi Mat. Nauk 25 (1970), no. 3 (153), 132-138. (Russian) MR 0280985 (43:6704)
  • [22] A. Pełczyński, Some problems on bases in Banach and Fréchet spaces, Israel J. Math. 2 (1964), 132-138. MR 30 #3356. MR 0173141 (30:3356)
  • [23] W. Ruckle, Representation and series summability of complete biorthogonal sequences, Pacific J. Math. 34 (1970), 511-528. MR 42 #2219. MR 0267317 (42:2219)
  • [24] I. Singer, On the constants of basic sequences in Banach spaces, Studia Math. 31 (1968), 125-134. MR 38 #6338. MR 0238062 (38:6338)
  • [25] -, Bases in Banach spaces. II, Stud. i Cerc. Mat. 15 (1964), 157-208. (Romanian) MR 32 #8121b.
  • [26] -, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), 183-188. MR 0350387 (50:2880)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0317011-5
Keywords: Basis, Markushevich basis, reflexivity, projection, complemented subspace, somewhat reflexive
Article copyright: © Copyright 1973 American Mathematical Society

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