Abstract
In any earnest treatment of sequences and series in Banach spaces a featured role must be reserved for basic sequences. Our initial discussion of this important notion will occupy this whole chapter. A foundation will be laid on which we will build several of the more interesting constructs in the theory of sequences and series in Banach spaces.
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Bibliography
Alaoglu, L. 1940. Weak topologies in nonned linear spaces. Ann. Math. 41, 252 – 267.
Bessaga, G. and Pelczynski, A. 1958. On bases and unconditional convergence of series in Banach spaces. Studia Math., 17, 151 – 164.
Botschkariev, S. V. 1974. Existence of a basis in the space of analytic functions, and some properties of the Franklin system. Mat. Sb., 24, 1 – 16.
Carleson, L. An explicit unconditional basis in H1.
Day, M. M. 1962. On the basis problem in normed spaces. Proc. Amer. Math. Soc., 13, 655 – 658.
Enflo, P. 1973. A counterexample to the approximation property in Banach spaces. Acta Math., 130, 309 – 317.
Enflo, P. Banach space with basis without a normalized monotone basis. Ark. Math.
Gelbaum, B. R. 1958. Banach spaces and bases. An. Acad. Brasil. Ci., 30, 29 – 36.
James, R. C. 1950. Bases and reflexivity of Banach spaces. Ann. of Math., 52, 518 – 527.
James, R. C. 1951. A non-reflexive Banach space isometric to its second conjugate space. Proc. Nat. Acad. Sci. (USA), 37, 174 – 177.
James, R. C. 1982. Bases in Banach spaces. Amer. Math. Mon., 89, 625 – 640.
Johnson, W. B. and Rosenthal, H. P. 1972. On w* basic sequences and their applications to the study of Banach spaces. Studia Math., 43, 77 – 92.
Kadec, M. I. and Pelczynski, A. 1965. Basic sequences, biorthogonal systems and norming sets in Banach and Frechet spaces. Studia Math., 25, 297 – 323.
Marcinkiewicz, J. 1937. Quelques théormès sur les séries orthogonales. Ann. Soc. Polon. Math., 16, 84 – 96.
Markusevich, A. I. 1943. On a basis in the wide sense for linear spaces. Dokl. Akad Nauk. SSSR, 41, 241–244.
Maurey, B. 1981. Isomorphismes entre espaces H1. Acta Math., 145, 79 – 120.
Ovsepian, R. I. and Pelczynski, A. 1975. The existence in every separable Banach space of a fundamental total and bounded biorthogonal sequence and related constructions of uniformly bounded orthonormal systems in L2. Studia Math., 54, 149 – 159.
Pelczynski, A. 1962. A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces.” Studia Math., 21, 371 – 374.
Pelczynski, A. 1976. All separable Banach spaces admit for every ε > 0 fundamental and total biorthogonal sequences bounded by 1 + ε. Studia Math., 55, 295 – 304.
Schauder, J. 1927. Zur Theorie stetiger Abbildungen in Funktionalräumen. Math. Zeit., 26, 47 – 65.
Schauder, J. 1928. Eine Eigenschaft des Haarschen Orthogonalsystems. Math. Z., 28, 317 – 320.
Wojtasczyck, P. The Franklin system is an unconditional basis in H1
Zippin, M. 1968. A remark on bases and reflexivity in Banach spaces. Israel J. Math., 6, 74 – 79.
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© 1984 Springer-Verlag New York, Inc.
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Diestel, J. (1984). Basic Sequences. In: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol 92. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5200-9_5
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DOI: https://doi.org/10.1007/978-1-4612-5200-9_5
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