Book Review
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MathSciNet review:
1568071
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Book Information:
Author:
Sylvie Guerre-Delabri\`ere
Title:
Classical sequences in Banach spaces
Additional book information:
Marcel Dekker, Inc., New York, 1992, xiv + 207 pp., US$99.75. ISBN 0-8247-8723-4.
David J. Aldous, Subspaces of $L^{1}$, via random measures, Trans. Amer. Math. Soc. 267 (1981), no. 2, 445–463. MR 626483, DOI 10.1090/S0002-9947-1981-0626483-2
[2] S. Banach, Théorie des opérations linéaires, Warszawa, 1932.
C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
Antoine Brunel and Louis Sucheston, On $J$-convexity and some ergodic super-properties of Banach spaces, Trans. Amer. Math. Soc. 204 (1975), 79–90. MR 380361, DOI 10.1090/S0002-9947-1975-0380361-2
D. L. Burkholder, A nonlinear partial differential equation and the unconditional constant of the Haar system in $L^{p}$, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 591–595. MR 670133, DOI 10.1090/S0273-0979-1982-15061-3
D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
Peter G. Casazza, Some questions arising from the homogeneous Banach space problem, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 35–52. MR 1209445, DOI 10.1090/conm/144/1209445
Peter G. Casazza and Thaddeus J. Shura, Tsirel′son’s space, Lecture Notes in Mathematics, vol. 1363, Springer-Verlag, Berlin, 1989. With an appendix by J. Baker, O. Slotterbeck and R. Aron. MR 981801, DOI 10.1007/BFb0085267
Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no $l_{p}$, Compositio Math. 29 (1974), 179–190. MR 355537
[12] W. T. Gowers, On the hyperplane problem, preprint.
[13] -, A Banach space not containing
,
or a reflexive subspace, preprint.
[14] -, A new dichotomy for Banach spaces, preprint.
[15] -, A solution to the Schroeder-Bernstein problem for Banach spaces, preprint.
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
S. Guerre and M. Levy, Espaces $l^{p}$ dans les sous-espaces de $L^{1}$, Trans. Amer. Math. Soc. 279 (1983), no. 2, 611–616 (French, with English summary). MR 709571, DOI 10.1090/S0002-9947-1983-0709571-3
Robert C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. MR 39915, DOI 10.2307/1969430
Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI 10.2307/1970663
Robert C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738–743. MR 417763, DOI 10.1090/S0002-9904-1974-13580-9
M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
Ryszard A. Komorowski and Nicole Tomczak-Jaegermann, Banach spaces without local unconditional structure, Israel J. Math. 89 (1995), no. 1-3, 205–226. MR 1324462, DOI 10.1007/BF02808201
J. L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), no. 1, 1–29. MR 407568, DOI 10.2307/1971054
J.-L. Krivine and B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (1981), no. 4, 273–295 (French, with English summary). MR 636897, DOI 10.1007/BF02761674
J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 263–269. MR 276734, DOI 10.1007/BF02771592
J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971), 379–390. MR 313780, DOI 10.1007/BF02771656
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
R. Daniel Mauldin (ed.), The Scottish Book, Birkhäuser, Boston, Mass., 1981. Mathematics from the Scottish Café; Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979. MR 666400
Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90
Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
Edward Odell and Thomas Schlumprecht, The distortion problem, Acta Math. 173 (1994), no. 2, 259–281. MR 1301394, DOI 10.1007/BF02398436
A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. MR 126145, DOI 10.4064/sm-19-2-209-228
Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 358307, DOI 10.1073/pnas.71.6.2411
Haskell P. Rosenthal, On subspaces of $L^{p}$, Ann. of Math. (2) 97 (1973), 344–373. MR 312222, DOI 10.2307/1970850
[36] -, Characterization of Banach spaces containing
, preprint.
Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. MR 1177333, DOI 10.1007/BF02782845
[38] -, A complementary minimal Banach space not containing
or
, preprint.
[39] B. S. Tsirelson, Not every Banach space contains
or
, Functional Anal. Appl. 8 (1974), 138-141.
- [1]
- D. Aldous, Subspaces of
via random measures, Trans. Amer. Math. Soc. 267 (1981), 445-463. MR 626483 (83b:46025)
- [2]
- S. Banach, Théorie des opérations linéaires, Warszawa, 1932.
- [3]
- C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 0115069 (22:5872)
- [4]
- A. Brunel and L. Sucheston, J-convexity and ergotic super-properties of Banach spaces, Trans. Amer. Math. Soc. 204 (1975), 79-90. MR 0380361 (52:1261)
- [5]
- D. L. Burkholder, A nonlinear partial differential equation and the unconditional constant of the Haar system in
, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 591-595. MR 670133 (83j:60045)
- [6]
- -, Boundary value problems and sharp inequalities for martingale transforms, Ann. of Probab. 12 (1984), 647-701. MR 744226 (86b:60080)
- [7]
- P. G. Casazza, Some questions arising from the homogeneous Banach space problem, Banach Spaces (B-L. Lin and W. B. Johnson, eds.) Contemp. Math. vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 35-52. MR 1209445 (93j:46016)
- [8]
- P. G. Casazza and T. Shura, Tsirelson's space, Lecture Notes in Math., vol. 1363, Springer-Verlag, New York, 1989. MR 981801 (90b:46030)
- [9]
- A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Sympos. on Linear Spaces, Jerusalem, 1961, pp. 123-160. MR 0139079 (25:2518)
- [10]
- P. Enflo, A counterexample to the approximation property in Banach spaces, Acta Math. 130 (1973), 309-317. MR 0402468 (53:6288)
- [11]
- T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no
, Composito Math. 29 (1974), 179-190. MR 0355537 (50:8011)
- [12]
- W. T. Gowers, On the hyperplane problem, preprint.
- [13]
- -, A Banach space not containing
,
or a reflexive subspace, preprint.
- [14]
- -, A new dichotomy for Banach spaces, preprint.
- [15]
- -, A solution to the Schroeder-Bernstein problem for Banach spaces, preprint.
- [16]
- W. T. Gowers and B. Maurey, The unconditional basic sequence problem, preprint. MR 1201238 (94k:46021)
- [17]
- A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc., vol. 16, Amer. Math. Soc., Providence, RI, 1955. MR 0075539 (17:763c)
- [18]
- S. Guerre and M. Levy, Espaces
dans les sons-espaces de
, Trans. Amer. Math. Soc. 279 (1983), 611-616. MR 709571 (85d:46033)
- [19]
- R. C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518-527. MR 0039915 (12:616b)
- [20]
- -, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542-550. MR 0173932 (30:4139)
- [21]
- -, A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. (N.S.) 80 (1974), 738-743. MR 0417763 (54:5811)
- [22]
- M. I. Kadec and A. Pelczynski, Bases, lacunary sequences, and complemented subspaces of the space
, Studia Math. 21 (1962), 161-176. MR 0152879 (27:2851)
- [23]
- R. Komorowski and N. Tomczak-Jaegermann, Banach spaces without local unconditional structure, preprint. MR 1324462 (96g:46007)
- [24]
- J. L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), 1-29. MR 0407568 (53:11341)
- [25]
- J. L. Krivine and B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (1981), 273-295. MR 636897 (83a:46030)
- [26]
- J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1972), 263-269. MR 0276734 (43:2474)
- [27]
- -, On Orlicz sequence spaces, Israel J. Math. 10 (1971), 379-390. MR 0313780 (47:2334)
- [28]
- -, Classical Banach spaces, II: function spaces, Ergeb. Math. Grenzgeb., vol. 92, Springer-Verlag, New York, 1979. MR 540367 (81c:46001)
- [29]
- R. Mauldin (editor), The Scottish book; mathematics from the Scottish Café, Birkhäuser, Boston, MA, 1981. MR 666400 (84m:00015)
- [30]
- B. Maurey and G. Pisier, Séries de variables aléatoires vectorièlles independantes et priopriétés géométriques des espaces de Banach, Studia Math 58 (1976), 45-90. MR 0443015 (56:1388)
- [31]
- V. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Math., vol. 1200, Springer-Verlag, New York, 1986. MR 856576 (87m:46038)
- [32]
- E. Odell and T. Schlumprecht, The distortion problem, preprint. MR 1301394 (96a:46031)
- [33]
- A. Pelczynski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228. MR 0126145 (23:A3441)
- [34]
- H. P. Rosenthal, A characterization of Banach spaces containing
, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. MR 0358307 (50:10773)
- [35]
- -, On subspaces of
, Ann. of Math. (2) 97 (1973), 344-373. MR 0312222 (47:784)
- [36]
- -, Characterization of Banach spaces containing
, preprint.
- [37]
- T. Schlumprecht, An arbitrarily distortable Banach space, preprint. MR 1177333 (93h:46023)
- [38]
- -, A complementary minimal Banach space not containing
or
, preprint.
- [39]
- B. S. Tsirelson, Not every Banach space contains
or
, Functional Anal. Appl. 8 (1974), 138-141.
Review Information:
Reviewer:
Peter G. Casazza
Journal:
Bull. Amer. Math. Soc.
30 (1994), 117-124
DOI:
https://doi.org/10.1090/S0273-0979-1994-00444-6