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On the theory of fundamental norming bounded biorthogonal systems in Banach spaces


Author: Paolo Terenzi
Journal: Trans. Amer. Math. Soc. 299 (1987), 497-511
MSC: Primary 46B15
DOI: https://doi.org/10.1090/S0002-9947-1987-0869217-4
MathSciNet review: 869217
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Abstract: Let $ X$ and $ Y$ be quasi complementary subspaces of a separable Banach space $ B$ and let $ ({z_n})$ be a sequence complete in $ X$. Then

(a) there exists a uniformly minimal norming $ M$-basis $ ({x_n})$ of $ X$ with $ {x_m} \in \operatorname{span} {({z_n})_{n \geqslant {q_m}}}$ for every $ m$, $ {q_m} \to \infty $;

(b) if $ ({x_n})$ is a uniformly minimal norming $ M$-basis of $ X$, there exists a uniformly minimal norming $ M$-basis of $ B$ which is an extension of $ ({x_n})$;

(c) there exists a uniformly minimal norming $ M$-basis $ ({x_n}) \cup ({y_n})$ of $ B$ with $ ({x_n}) \subset X$ and $ ({y_n}) \subset Y$.


References [Enhancements On Off] (What's this?)

  • [1] S. Banach, Théorie des opérations linéaires, Chelsea, New York, 1932.
  • [2] W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. 45 (1973), 173-179. MR 0331021 (48:9356)
  • [3] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309-317. MR 0402468 (53:6288)
  • [4] M. A. Krasnoselskii, M. G. Krein and D. P. Milman, On defect numbers of linear operators in a Banach space and on some geometric problems, Sb. Trud. Inst. Matem. Akad. Nauk Ukr. SSR 11 (1948), 97-112.
  • [5] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin and New York, 1977. MR 0500056 (58:17766)
  • [6] G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), 322-325. MR 0015676 (7:455c)
  • [7] A. Markushevich, Sur les bases (au sens large) dans les espèces linéaires, Dokl. Akad. Nauk SSSR 41 (1943), 227-229. MR 0010778 (6:69b)
  • [8] V. D. Milman, Geometric theory of Banach spaces. I, Russian Math. Surveys 25 (1970), 111-170. MR 0280985 (43:6704)
  • [9] R. I. Ovsepian and A. Pełczynski, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $ {L^2}$, Studia Math. 54 (1975), 149-159. MR 0394137 (52:14942)
  • [10] A. Pelczynski, All separable Banach spaces admit for every $ \varepsilon > 0$ fundamental total and bounded by $ 1 + \varepsilon $ biorthogonal sequences, Studia Math. 55 (1976), 295-304. MR 0425587 (54:13541)
  • [11] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, Berlin and New York, 1971. MR 0270044 (42:4937)
  • [12] -, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), 183-188. MR 0350387 (50:2880)
  • [13] -, Bases in Banach spaces. II, Springer-Verlag, Berlin and New York, 1981. MR 610799 (82k:46024)
  • [14] A. Sobczyk, Projection of the space $ m$ on its subspace $ {c_0}$, Bull. Amer. Math. Soc. 47 (1941), 938-947. MR 0005777 (3:205f)
  • [15] P. Terenzi, Properties of structure and completeness, in a Banach space, of the sequences without an infinite minimal subsequence, 1st. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), 42-66. MR 555283 (81e:46014)
  • [16] -, Biorthogonal systems in Banach spaces, Riv. Mat. Univ. Parma (4) 4 (1978), 165-204. MR 540692 (80g:46016)
  • [17] -, Some completeness properties of general sequences in a Banach space, Bollettino UMI 5 (15-B) (1978), 743-753. MR 524096 (80d:46026)
  • [18] -, On bounded and total biorthogonal systems spanning given subspaces, Accad. Naz. dei Lincei (Rend. Sc.) 58 (1979), 168-178. MR 622788 (82g:46030)
  • [19] -, Extension of uniformly minimal $ M$-basic sequences in Banach spaces, J. London Math. Soc. 27 (1983), 500-506. MR 697142 (85b:46016)
  • [20] W. A. Veech, Short proof of Sobczyk's theorem, Proc. Amer. Math. Soc. 28 (1971), 627-628. MR 0275122 (43:879)
  • [21] V. S. Vinokurov, On biorthogonal systems spanning a given subspace, Dokl. Akad. Nauk SSSR 85 (1952), 685-689. MR 0049480 (14:183b)
  • [22] A. Wilansky, Functional analysis, Blaisdell, Waltham, Mass., 1964. MR 0170186 (30:425)
  • [23] -, Modern methods in topological vector spaces, McGraw-Hill, New York, 1978.
  • [24] M. Zippin, The separable extension problem, Israel J. Math. 26 (1977), 372-387. MR 0442649 (56:1030)

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DOI: https://doi.org/10.1090/S0002-9947-1987-0869217-4
Article copyright: © Copyright 1987 American Mathematical Society

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