Chapter 23 Descriptive set theory and Banach spaces

Handbook of the Geometry of Banach Spaces

Volumn 2, Issue , 2003, Pages 1007-1069

  Argyros, Spiros A ^a   Godefroy, Gilles ^b   Rosenthal, Haskell P ^c  

^a UNIVERSITY OF ATHENS   (Greece)

^b SORBONNE UNIVERSITÉ   (France)

^c UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 24644441138     PISSN: 18745849     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5849(03)80030-X     Document Type: Chapter

References (133)

1. Alspach D., and Argyros S.A. Complexity of weakly null sequences. Dissertationes Math. 321 (1992) 1-44
2. p, 1 < p < ∞, p ≠ 2. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 1 (2001), Elsevier, Amsterdam 123-160
3. Androulakis G., and Odell E. Distorting mixed Tsirelson spaces. Israel J. Math. 109 (1999) 125-149
4. Argyros S.A. A universal property of hereditarily indecomposable Banach spaces. Proc. Amer. Math. Soc. 129 (2001) 3231-3239
5. 1 Banach spaces. Trans. Amer. Math. Soc. 349 (1997) 973-995
6. Argyros S.A., Deliyanni I., Kutzarova D., and Manoussakis A. Modified mixed Tsirelson spaces. J. Funct. Anal. 159 (1998) 43-109
7. S.A. Argyros, I. Deliyanni and A. Manoussakis, Distortion and spreading models in modified mixed Tsirelson spaces, Studia Math., to appear.
8. Argyros S.A., and Felouzis V. Interpolating hereditarily indecomposable Banach spaces. J. Amer. Math.Soc. 13 2 (2000) 243-294
9. Argyros S.A., and Gasparis I. Unconditional structures of weakly null sequences. Trans. Amer. Math. Soc. 353 (2001) 2019-2058
10. Argyros S.A., and Kanellopoulos V. Optimal sequences of continuous functions converging to a Baire-1 function. Math. Ann. 324 (2002) 689-729
11. Argyros S.A., Mercourakis S., and Tsarpalias A. Convex unconditionality and summability of weakly null sequences. Israel J. Math. 107 (1998) 157-193
12. S.A. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Preprint.
13. Baire R. Sur les fonctions de variables réelles. Ann. Mat. Pura Applicata Ser. 1113 (1899) 1-122
14. p. J. Funct. Anal. 69 2 (1986) 207-228
15. Bernués J., and Deliyanni I. Families of finite subsets of of low complexity and Tsirelson type spaces. Math. Nachr. 222 (2001) 15-29
16. Bessaga C., and Pelczyński A. On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958) 151-164
17. Bossard B. Codages des espaces de Banach séparables. Familles analytiques ou coanalytiques d'espaces de Banach. Notes aux CRAS Paris 316 1 (1993) 1005-1010
18. B-Bossard. An ordinal version of some applications of the classical interpolation theorem. Fund. Math. 152 1 (1997) 55-74
19. Bossard B. On a problem of H.P. Rosenthal. Houston J. Math. 26 1 (2000) 1-16
20. Bossard B. A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces. Fund. Math. 172 (2002) 117-152
21. Bossard B., and López G. The point of continuity property descriptive complexity and ordinal index. Serdica Math. J. 24 (1998) 199-214
22. Bourgain J. The Szlenk index operators on C (K)-spaces. Bull. Soc. Math. Belg. Sér. B 31 (1979) 87-117
23. Bourgain J. On separable Banach spaces universal for all separable reflexive spaces. Proc. Amer. Math. Soc. 79 (1980) 241-246
24. Bourgain J. On convergent sequences of continuous functions. Bull. Soc. Math. Belg. Sér. B 32 (1980) 235-249
25. Bourgain J., Fremlin D., and Talagrand M. Pointwise compact sets of Baire-measurable functions. Amer. J. Math. 100 4 (1978) 845-886
26. p. Ann. of Math. 114 (1981) 803-831
27. Brunei A., and Sucheston L. On B-convex Banach spaces. Math. Systems Theory 7 (1974) 294-299
28. Brunei A., and Sucheston L. On J-convexity and some ergodic superproperties of Banach spaces. Trans. Amer. Math. Soc. 204 (1975) 79-90
29. (French). Capon M. Sur les fonctions qui vérifient le calcul barycentrique. Proc. London Math. Soc. (3) 32 1 (1976) 163-180
30. 83. Casazza P.G., and Odell E. Tsirelson space and minimal subspaces. 61-72 (1982), Longhorn Notes University of Texas
31. P.G. Casazza and T. Shura, Tsirelson's Space, Lecture Notes in Math. 1363, Springer, New York.
32. Chaatit F. On simple DBSC functions. Math. Japonica 46 2 (1997) 357-361
33. Chaatit F., Mascioni V., and Rosenthal H.P. On functions of finite Baire index. J. Funct. Anal. 142 (1996) 277-295
34. Chaatit F., and Rosenthal H.P. On differences of semi-continuous functions. Quaestiones Math. 23 (2000) 295-311
35. 13 pp. Choquet G. Remarques à propos de la démonstration de l'unicité de P.A. Meyer. Seminaire Brelot-Choquet-Deny Théorie du Potentiel 6 8 (1962)
36. Christensen J.R.R. Topology Borel Structure. North-Holland Math. Stud. 10 (1974)
37. Davis W.B., and Johnson W.B. On the existence of fundamental and total biorthogonal systems in Banach spaces. Studia Math. 45 (1973) 173-179
38. Debs G. Effective properties in compact sets of Borel functions. Mathematika 34 (1987) 64-68
39. Ellentuck E. A new proof that analytic sets are Ramsey. J. Symbolic Logic 39 (1974) 163-165
40. Elton J. Weakly null normalized sequences in Banach spaces. Doctoral thesis (1978), Yale University
41. Erdös P., and Magidor M. A note on regular methods of summability and the Banach-Saks property. Proc. Amer. Math. Soc. 59 (1976) 232-234
42. Farmaki V. Characterizations of elements of a double dual Banach space their canonical reproduc-tions. Studia Math. 104 1 (1993) 61-74
43. Farmaki V. On Baire-1/ 4 functions spreading models. Mathematika 41 (1994) 251-265
44. 0-spreading models. Trans. Amer. Math. Soc. 345 (1994) 819-831
45. Farmaki V. On Baire-1/4 functions. Trans. Amer. Math. Soc. 348 (1996) 4023-4041
46. 0 sequences in Banach spaces. Studia Math. 135 (1999) 83-102
47. 0-content and semibounded complete-ness. Fund. Math. 172 (2002) 153-179
48. 1-behavior of a sequence of functions, Positivity, to appear.
49. p. Compositio Math. 29 (1974) 179-190
50. Finet C., and Godefroy G. Biorthogonal systems and big quotient spaces. In: Lin B.-L. (Ed). Banach Space Theory. Contemp. Math. 85 (1989) 87-109
51. Fonf V.R. One property of Lindenstrauss-Phelps spaces. Functional Anal. Appl. 13 (1979) 66-67
52. Fonf V., Lindenstrauss J., and Phelps R.R. Infinite dimensional convexity. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 1 (2001), Elsevier, Amsterdam 599-670
53. Gasparis I. A dichotomy theorem for subsets of the power set of the natural numbers. Proc. Amer. Math. Soc. 129 (2001) 759-764
54. I. Gasparis, Strictly singular non-compact operators on hereditarily indecomposable Banach spaces, Proc. Amer. Math. Soc, to appear.
55. δ-embeddings in Hilbert space. J. Funct. Anal. 61 (1985) 72-97
56. Godefroy G. Compacts de Rosenthal. Pacific J. Math. 91 (1980) 293-306
57. Godefroy G. The Szlenk index and its applications. In: Banakh T. (Ed). General Topology in Banach Spaces (2001), Nova Sci. Publ. Huntington, New York 71-79
58. Godefroy G. Renormings of Banach spaces. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 1 (2001), Elsevier, Amsterdam 781-836
59. Godefroy G., and Li D. Banach spaces which are M-ideals in their bidual have property (u). Ann. Inst. Fourier 39 (1989) 361-371
60. Godefroy G., and Louveau A. Axioms of determinacy and biorthogonal systems. Israel J. Math. 67 1 (1989) 109-116
61. Godefroy G., and Talagrand M. Espaces de Banach représentables. Israel J. Math. 41 4 (1982) 321-330
62. (this Handbook). Gowers W.T. Ramsey methods in Banach spaces. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 2 (2003), Elsevier, Amsterdam 1071-1097
63. Gowers W.T., and Maurey B. The unconditional basic sequence problem. J. Amer. Math. Soc. 6 (1993) 851-874
64. Grothendieck A. Sur les applications linéaires faiblement compactes d'espaces du type C (K). Canad. J. Math. 5 (1953) 129-173
65. Springer. Hammand P., Werner D., and Werner W. M-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Math. 1547 (1993)
66. Springer. Haydon R., Odell E., and Rosenthal H. On certain classes of Baire-1 functions with applications to Banach space theory. Functional Analysis, Lecture Notes in Math. 1470 (1991) 1-35
67. Hurewicz W. Zur Theorie der analytischen Mengen. Fund. Math. 15 (1930) 4-16
68. James R.C. Uniformly non square Banach spaces. Ann. of Math. 80 (1964) 542-550
69. Johnson W.B. A reflexive Banach space which is not sufficiently Euclidean. Studia Math. 55 (1976) 201-205
70. Kaufman R. PCA sets and convexity. Fund. Math. 163 (2000) 267-275
71. Kaufman R. On smooth norms and analytic sets. Israel J. Math. 116 (2000) 21-27
72. Kechris A. Classical Descriptive Set Theory (1995), Springer, New York
73. Kechris A.S., and Louveau A. Descriptive Set Theory and the Structure of Sets of Uniqueness. London Math. Soc. Lecture Note Series 128 (1987)
74. Kechris A.S., and Louveau A. A classification of Baire class 1 functions. Trans. Amer. Math. Soc. 318 (1990) 209-236
75. Kiriakouli P. A classification of Baire-1 functions. Trans. Amer. Math. Soc. 351 (1999) 4599-4609
76. Kiriakouli P. Characterizations of spreading models of 1. Comment. Math. Univ. Carolin. 41 (2000) 79-95
77. P. Kiriakouli and S. Negrepontis, Baire-1 functions and spreading models of Preprint.
78. Komisarski A. Some descriptive set-theoretic properties of the isomorphism relation between Banach spaces. Proc. Amer. Math. Soc. 129 (2001) 3085-3090
79. Kuratowski K. Topology Vol. I (1966), Academic Press, New York
80. Kutzarova D.N., and Lin P.K. Remarks about Schlumprecht space. Proc. Amer. Math. Soc. 128 (2000) 2059-2068
81. Landen G. Dentability indices and locally uniformly convex renormings. Rocky Mountain J. Math. 23 2 (1993) 635-647
82. Landen G. On the Szlenk index and the weak*-dentability index. Quart. J. Math. Oxford 47 2 (1996) 59-71
83. Lee P.Y., Tang W.-K., and Zhao D. An equivalent definition of functions of the first Baire class. Proc. Amer. Math. Soc. 129 (2001) 2273-2275
84. D. Leung and W.-K. Tang, Functions of Baire class one. Preprint.
85. López G. Banach spaces with supershrinking basis. Studia Math. 132 (1999) 29-36
86. López G., Martin M., and Payá R. Real Banach spaces with numerical index 1. Bull. London Math. Soc. 31 (1999) 207-212
87. Lusin N.N. Sur la classification de M. Baire. C.R. Acad. Sci. Paris 141 (1917) 91-94
88. Lusin N., and Sierpinski W. Sur quelques propriétés des ensembles (A). Bull. Int. Acad. Sci. Cracovie Sér. A (1918) 35-48
89. Manoussakis A. On the structure of a certain class of mixed Tsirelson spaces. Positivity 5 (2001) 193-238
90. 1 Banach spaces, Proc. Amer. Math. Soc, to appear.
91. Marciszewski W. On a classification of pointwise compact sets of the first Baire class functions. Fund.Math. 133 (1989) 195-209
92. (this Handbook). Maurey B. Banach spaces with few operators. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 2 (2003), Elsevier, Amsterdam
93. Maurey B., and Rosenthal H.P. Normalized weakly null sequences with no unconditional subsequences. Studia Math. 61 (1977) 77-98
94. McWilliams R.D. Iterated ω*-sequential closure of a Banach space in its second conjugate. Proc. Amer. Math. Soc. 16 (1965) 1195-1199
95. Mercourakis S. On Cesaro summable sequences of continuous functions. Mathematika 42 (1995) 87-104
96. Mercourakis S., and Negrepontis S. Banach spaces and topology II. In: Huskek M., and Mill J.v. (Eds). Recent Progress in General Topology (1992), North-Holland, Amsterdam 493-536
97. p spaces and bounded distortions. Contemp. Math. 144 (1993) 173-195
98. Moschovakis Y.N. Descriptive Set Theory (1980), North-Holland, Amsterdam
99. Odell E. Applications of Ramsey theorems to Banach space theory. In: Lacey H.E. (Ed). Notes in Banach Space (1980), University of Texas Press 379-404
100. Odell E. On Schreier unconditional sequences. Contemp. Math. 144 (1993) 197-201
101. E. Odell, On Subspaces, Asymptotic Structure and Distortion of Banach spaces, Connections with Logic, Analysis and Logic, Part Three, LMS Cambridge University Press.
102. 1. Israel J. Math. 20 (1975) 375-384
103. Odell E., and Schlumprecht Th. A problem on spreading models. J. Funct. Anal. 153 (1998) 249-261
104. (this Handbook). Odell E., and Schlumprecht Th. Distortion and asymptotic structure. In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 2 (2003), Elsevier, Amsterdam
105. 1 spaces. J. Funct. Anal. 150 1 (1997) 101-145
106. Pelczyński A. A connection between weakly unconditional convergence weak completeness of Banach spaces. Bull. Acad. Polon. Sci. 6 (1958) 251-253
107. Pelczyński A. Universal bases. Studia Math. 32 (1969) 247-268
108. Phelps R. Lectures on Choquet's Theorem (1966), D. van Nostrand Co., Princeton, NJ
109. Pol R. Note on pointwise convergence of sequences of analytic sets. Mathematika 36 (1989) 290-300
110. Ptak V. A combinatorial lemma on the existence of convex means and its application to weak compactness. In: Klee C.V. (Ed). Proc. Symp. Pure Math VII (1963), Amer. Math. Soc, Providence
111. (French). Rogalski M. Opérateurs de Lion projecteurs boréliens et simplexes analytiques. J. Funct. Anal. 2 (1968) 458-488
112. 1. Proc. Nat. Acad. Sci. USA 71 (1974) 2411-2413
113. Rosenthal H.P. Weakly independent sequences and the Banach-Saks property. Bull. London Math. Soc. 8 (1976) 22-24
114. Rosenthal H.P. Point-wise compact subsets of the first Baire class. Amer. J. Math. 99 (1977) 362-378
115. Rosenthal H.P. Some recent discoveries in the isomorphic theory of Banach spaces. Bull. Amer. Math. Soc. 84 5 (1978) 803-831
116. c, Paris VI 1978-79 (1979) 5
117. Rosenthal H.P. Weak*-Polish Banach spaces. J. Funct. Anal. 76 (1988) 267-316
118. 0. J. Amer. Math. Soc. 7 (1994) 707-748
119. H.P. Rosenthal, Differences of bounded semi-continuous functions I, to appear.
120. (this Handbook). Rosenthal H.P. The Banach spaces C(K). In: Johnson W.B., and Lindenstrauss J. (Eds). Handbook of the Geometry of Banach Spaces Vol. 2 (2003), Elsevier, Amsterdam
121. Schlumprecht Th. An arbitrarily distortable Banach space. Israel J. Math. 76 (1991) 81-95
122. Schreier J. Ein gegenbeispiel zur theorie der schwachen konvergence. Studia Math. 2 (1930) 58-62
123. Sersouri A. A note on the Lavrientiev index for quasi-reflexive Banach spaces. Contemp. Math. 85 (1989) 497-508
124. Silver J. Every analytic set is Ramsey. J. Symbolic Logic 35 (1970) 60-64
125. Singer I. On biorthogonal systems total sequences offunctionals II. Math. Ann. 201 (1973) 1-8
126. Souslin M.M. Sur une definition des ensembles mesurables B sans nombres transfinis. C.R. Acad. Sci. Paris 141 (1917), 88-91
127. Stegall C. The Radon-Nikodym property in conjugate Banach spaces. Trans. Amer. Math. Soc. 206 (1975) 213-223
128. Szlenk W. The non existence of a separable reflexive Banach space universal for all separable Banach spaces. Studia Math. 30 (1968) 53-61
129. Talagrand M. A new type of affine Borel function. Math. Scand. 54 (1984) 183-188
130. Todorcević S. Compact subsets of the first Baire class. J. Amer. Math. Soc. 12 4 (1999) 1179-1212
131. Tomczak-Jaegermann N. Banach spaces of type p have arbitrarily distortable subspaces. GAFA 6 (1996) 1074-1082
132. 0. Functional Anal. Appl. 8 (1974) 138-141
133. Zinnmeister M. Les dérivations analytiques. Publ. Math. Univ. Paris-Sud 86T10 (1986)