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Complexity of weakly null sequences

Seria
Rozprawy Matematyczne tom/nr w serii: 321 wydano: 1992
Zawartość
Warianty tytułu
Abstrakty
EN
We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each α < ω₁, a weakly null sequence $(x^α_n)_n$ in $C(ω^{ω^{α}})$ with complexity α. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are used to construct Tsirelson-like spaces of large index. We also show that this new ordinal index is related to the Lavrent'ev index of a Baire-1 function and use the index to sharpen some results of Alspach and Odell on averaging weakly null sequences.
EN

CONTENTS
0. Introduction.................................................................................................5
1. Preliminaries...............................................................................................6
2. Weakly null sequences and the l¹-index......................................................9
3. Comparison with the l¹-index.....................................................................12
4. Construction of weakly null sequences with large oscillation index............21
5. Reflexive spaces with large oscillation index.............................................33
6. Comparison with the averaging index........................................................37
References....................................................................................................43
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 321
Liczba stron
44
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXI
Daty
wydano
1992
otrzymano
1991-12-20
poprawiono
1992-03-16
Twórcy
  • Department of Mathematics Oklahoma State University Stillwater, OK, 74078-0613 U.S.A.
  • Department of Mathematics University of Crete Heraklion, Crete Greece
Bibliografia
  • [A-O] D. Alspach and E. Odell, Averaging Weakly Null Sequences, Lecture Notes in Math. 1332, Springer, Berlin 1988.
  • [A] S. Argyros, Banach spaces of the type of Tsirelson, preprint.
  • [Bo] J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belg. Sér. B 32 (1980), 235-249.
  • [C-S] P. Casazza and T. Shura, Tsirelson's Space, Lecture Notes in Math. 1363, Springer, Berlin 1989.
  • [D1] J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Math. 485, Springer, Berlin 1975.
  • [D2] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer, Berlin 1984.
  • [Dor] L. Dor, On projections in $L_1$, Ann. of Math. 102 (1975), 463-474.
  • [G-H] D. C. Gillespie and W. A. Hurwitz, On sequences of continuous functions having continuous limits, Trans. Amer. Math. Soc. 32 (1930), 527-543.
  • [H-O-R] R. Haydon, E. Odell and H. P. Rosenthal, On Certain Classes of Baire-1 Functions with Applications to Banach Space Theory, Lecture Notes in Math. 1470, Sprin- ger, Berlin 1991.
  • [J] R. C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743.
  • [K-L] A. S. Kechris and A. Louveau, A classification of Baire-1 functions, Trans. Amer. Math. Soc. 318 (1990), 209-236.
  • [K] K. Kuratowski, g Topology, I, Academic Press, New York 1966.
  • [L-S] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $l_1$ and whose duals are non-separable, Studia Math. 54 (1975), 81-105.
  • [L-T,I] J. Lindenstrauss and L. Tzafriri, g Classical Banach Spaces I: Sequence Spaces, Ergeb. Math. Grenzgeb. 92, Springer, Berlin 1977.
  • [L-T,II] J. Lindenstrauss and L. Tzafriri, g Classical Banach Spaces II: Function Spaces, Ergeb. Math. Grenzgeb. 97, Springer, Berlin 1979.
  • [M-R] B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), 77-98.
  • [O] E. Odell, A normalized weakly null sequence with no shrinking subsequence in a Banach space not containing $l_1$, Compositio Math. 41 (1980), 287-295.
  • [P-S] A. Pełczyński and W. Szlenk, An example of a non-shrinking basis, Rev. Roumaine Math. Pures Appl. 10 (1965), 961-966.
  • [R] H. P. Rosenthal, A characterization of Banach spaces containing $l^1$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413.
  • [Sch] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2 (1930), 58-62.
  • [Sz] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, ibid. 30 (1968), 53-61.
  • [Z] Z. Zalcwasser, Sur une propriété du champ des fonctions continues, ibid. 2 (1930), 63-67.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 46B20.
Identyfikator YADDA
bwmeta1.element.dl-catalog-6b371b16-cc33-4d17-befa-89bf4becbc48
Identyfikatory
ISBN
83-85116-59-1
ISSN
0012-3862
Kolekcja
DML-PL
Zawartość książki

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