Examples concerning heredity problems of WCG Banach spaces
Authors:
Spiros A. Argyros and Sophocles Mercourakis
Journal:
Proc. Amer. Math. Soc. 133 (2005), 773785
MSC (2000):
Primary 46B20, 46B26, 03E05
Published electronically:
August 20, 2004
MathSciNet review:
2113927
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We present two examples of WCG spaces that are not hereditarily WCG. The first is a space with an unconditional basis, and the second is a space such that is WCG and does not contain . The nonWCG subspace of has the additional property that is not WCG and is reflexive.
 [A]
S.A. Argyros, Weakly Lindelöf Determined Banach spaces not containing , (1993), preprint.
 [ACGJM]
Spiros
A. Argyros, Jesús
F. Castillo, Antonio
S. Granero, Mar
Jiménez, and José
P. Moreno, Complementation and embeddings of
𝑐₀(𝐼) in Banach spaces, Proc. London Math. Soc.
(3) 85 (2002), no. 3, 742–768. MR 1936819
(2003k:46022), http://dx.doi.org/10.1112/S0024611502013618
 [AL]
D.
Amir and J.
Lindenstrauss, The structure of weakly compact sets in Banach
spaces, Ann. of Math. (2) 88 (1968), 35–46. MR 0228983
(37 #4562)
 [AM]
S.
Argyros and S.
Mercourakis, On weakly Lindelöf Banach spaces, Rocky
Mountain J. Math. 23 (1993), no. 2, 395–446. MR 1226181
(94i:46016), http://dx.doi.org/10.1216/rmjm/1181072569
 [F]
Marián
J. Fabian, Gâteaux differentiability of convex functions and
topology, Canadian Mathematical Society Series of Monographs and
Advanced Texts, John Wiley & Sons Inc., New York, 1997. Weak Asplund
spaces; A WileyInterscience Publication. MR 1461271
(98h:46009)
 [F1]
M.
Fabián, Each weakly countably determined Asplund space
admits a Fréchet differentiable norm, Bull. Austral. Math. Soc.
36 (1987), no. 3, 367–374. MR 923819
(89c:46018), http://dx.doi.org/10.1017/S000497270000366X
 [H]
Richard
Haydon, Some more characterizations of Banach spaces containing
𝑙₁, Math. Proc. Cambridge Philos. Soc.
80 (1976), no. 2, 269–276. MR 0423047
(54 #11031)
 [HHZ]
P. Habala, P. Hajek, and V. Zizler, Introduction to Banach Spaces I, II Matfyzpress, Prague (1996).
 [HO]
J.
Hagler and E.
Odell, A Banach space not containing 𝑙₁ whose dual
ball is not weak* sequentially compact, Illinois J. Math.
22 (1978), no. 2, 290–294. MR 0482087
(58 #2174)
 [J]
Robert
C. James, A separable somewhat reflexive Banach
space with nonseparable dual, Bull. Amer. Math.
Soc. 80 (1974),
738–743. MR 0417763
(54 #5811), http://dx.doi.org/10.1090/S000299041974135809
 [JL]
W.
B. Johnson and J.
Lindenstrauss, Some remarks on weakly compactly generated Banach
spaces, Israel J. Math. 17 (1974), 219–230. MR 0417760
(54 #5808)
 [JZ]
K.
John and V.
Zizler, Smoothness and its equivalents in weakly compactly
generated Banach spaces, J. Functional Analysis 15
(1974), 1–11. MR 0417759
(54 #5807)
 [LS]
J.
Lindenstrauss and C.
Stegall, Examples of separable spaces which do not contain
ℓ₁ and whose duals are nonseparable, Studia Math.
54 (1975), no. 1, 81–105. MR 0390720
(52 #11543)
 [MS]
S. Mercourakis and E. Stamati, A new class of weakly analytic Banach spaces, Mathematika (to appear).
 [R]
Haskell
P. Rosenthal, The heredity problem for weakly compactly generated
Banach spaces, Compositio Math. 28 (1974),
83–111. MR
0417762 (54 #5810)
 [T]
Michel
Talagrand, Espaces de Banach faiblement
\cal𝐾analytiques, Ann. of Math. (2) 110
(1979), no. 3, 407–438 (French). MR 554378
(81a:46021), http://dx.doi.org/10.2307/1971232
 [Z]
V. Zizler, Nonseparable Banach Spaces, Handbook of the Geometry of Banach spaces W.B. Johnson, J. Lindenstrauss (eds.), NorthHolland, Vol. 2, ch. 41, (2003), 17431816.
 [A]
 S.A. Argyros, Weakly Lindelöf Determined Banach spaces not containing , (1993), preprint.
 [ACGJM]
 S.A. Argyros, J.F. Castillo, A.S. Granero, M. Jimenez and J.P. Moreno, Complementation and embeddings of in Banach spaces, Proc. London Math. Soc. (3) 85, (2002), 742768. MR 2003k:46022
 [AL]
 D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Annals of Math., 88, (1968), 3546. MR 37:4562
 [AM]
 S.A. Argyros and S. Mercourakis, On weakly Lindelöf Banach spaces, Rocky Mountain J. Math. 23, (1993), 395446. MR 94i:46016
 [F]
 M.J. Fabián, Gâteaux Differentiability of Convex Functions and Topology, John Wiley & Sons, Inc. (1997). MR 98h:46009
 [F1]
 M.J. Fabián, Each weakly countably determined Asplund space admits a Fréchet differentiable norm, Bull. Australian Math. Soc., 36, (1987), 367374. MR 89c:46018
 [H]
 R.G. Haydon, Some more characterizations of Banach spaces containing , Math. Proc. Cambridge Phil. Soc., 80, (1976), 269276. MR 54:11031
 [HHZ]
 P. Habala, P. Hajek, and V. Zizler, Introduction to Banach Spaces I, II Matfyzpress, Prague (1996).
 [HO]
 J. Hagler and E. Odell, A Banach space not containing whose dual ball is not weaksequentially compact, Illinois J. Math. 22, (1978), 290294. MR 58:2174
 [J]
 R.C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80, (1974), 738743. MR 54:5811
 [JL]
 W.B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17, (1974), 219230. MR 54:5808
 [JZ]
 K. John and V. Zizler, Smoothness and its equivalents in weakly compactly generated Banach spaces, J. Funct. Anal. 15, (1974), 161166. MR 54:5807
 [LS]
 J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain and whose duals are nonseparable, Studia Math. 54, (1975), 81105. MR 52:11543
 [MS]
 S. Mercourakis and E. Stamati, A new class of weakly analytic Banach spaces, Mathematika (to appear).
 [R]
 H.P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compos. Math. 28, (1974), 83111. MR 54:5810
 [T]
 M. Talagrand, Espaces de Banach faiblement analytiques, Ann. of Math. 110, (1979), 407438. MR 81a:46021
 [Z]
 V. Zizler, Nonseparable Banach Spaces, Handbook of the Geometry of Banach spaces W.B. Johnson, J. Lindenstrauss (eds.), NorthHolland, Vol. 2, ch. 41, (2003), 17431816.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
46B20,
46B26,
03E05
Retrieve articles in all journals
with MSC (2000):
46B20,
46B26,
03E05
Additional Information
Spiros A. Argyros
Affiliation:
Department of Mathematics, National Technical University of Athens, Athens 15780, Greece
Email:
sargyros@math.ntua.gr
Sophocles Mercourakis
Affiliation:
Department of Mathematics, University of Athens, Athens 15784, Greece
Email:
smercour@math.uoa.gr
DOI:
http://dx.doi.org/10.1090/S000299390407532X
PII:
S 00029939(04)07532X
Keywords:
WCG Banach space,
unconditional basis,
tree
Received by editor(s):
July 16, 2003
Received by editor(s) in revised form:
October 23, 2003
Published electronically:
August 20, 2004
Communicated by:
N. TomczakJaegermann
Article copyright:
© Copyright 2004 American Mathematical Society
