Separated sets and Auerbach systems in Banach spaces

Hájek, Petr; Kania, Tomasz; Russo, Tommaso

doi:10.1090/tran/8160

Separated sets and Auerbach systems in Banach spaces
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Trans. Amer. Math. Soc. 373 (2020), 6961-6998 Request permission

Abstract:

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains an uncountable $(1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $c_0(\omega _1)$. Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $(1+\varepsilon )$-separated subset of any regular cardinality not exceeding the density of $X$; should the space $X$ be super-reflexive, the unit sphere of $X$ contains such a subset of cardinality equal to the density of $X$. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

References

Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
Y. Benyamini and T. Starbird, Embedding weakly compact sets into Hilbert space, Israel J. Math. 23 (1976), no. 2, 137–141. MR 397372, DOI 10.1007/BF02756793
Paul Civin and Bertram Yood, Quasi-reflexive spaces, Proc. Amer. Math. Soc. 8 (1957), 906–911. MR 90020, DOI 10.1090/S0002-9939-1957-0090020-6
Marek Cúth, Ondřej Kurka, and Benjamin Vejnar, Large separated sets of unit vectors in Banach spaces of continuous functions, Colloq. Math. 157 (2019), no. 2, 173–187. MR 3984267, DOI 10.4064/cm7648-1-2019
Mahlon M. Day, On the basis problem in normed spaces, Proc. Amer. Math. Soc. 13 (1962), 655–658. MR 137987, DOI 10.1090/S0002-9939-1962-0137987-7
Sylvain Delpech, Separated sequences in asymptotically uniformly convex Banach spaces, Colloq. Math. 119 (2010), no. 1, 123–125. MR 2602060, DOI 10.4064/cm119-1-7
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
J. Elton and E. Odell, The unit ball of every infinite-dimensional normed linear space contains a $(1+\varepsilon )$-separated sequence, Colloq. Math. 44 (1981), no. 1, 105–109. MR 633103, DOI 10.4064/cm-44-1-105-109
Paul Erdős, András Hajnal, Attila Máté, and Richard Rado, Combinatorial set theory: partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, North-Holland Publishing Co., Amsterdam, 1984. MR 795592
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 Wiley-Interscience Publication. MR 1461271
Marián Fabián and Gilles Godefroy, The dual of every Asplund space admits a projectional resolution of the identity, Studia Math. 91 (1988), no. 2, 141–151. MR 985081, DOI 10.4064/sm-91-2-141-151
Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381, DOI 10.1007/978-1-4419-7515-7
Marián Fabian, Petr Hájek, and Václav Zizler, A note on lattice renormings, Comment. Math. Univ. Carolin. 38 (1997), no. 2, 263–272. MR 1455493
Catherine Finet and Gilles Godefroy, Biorthogonal systems and big quotient spaces, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 87–110. MR 983383, DOI 10.1090/conm/085/983383
E. Glakousakis and S. Mercourakis, On the existence of $1$-separated sequences on the unit ball of a finite-dimensional Banach space, Mathematika 61 (2015), no. 3, 547–558. MR 3415641, DOI 10.1112/S002557931400028X
G. Godefroy, N. J. Kalton, and G. Lancien, Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. 353 (2001), no. 10, 3895–3918. MR 1837213, DOI 10.1090/S0002-9947-01-02825-2
Gilles Godefroy and Michel Talagrand, Espaces de Banach représentables, Israel J. Math. 41 (1982), no. 4, 321–330 (French, with English summary). MR 657864, DOI 10.1007/BF02760538
B. V. Godun, A special class of Banach spaces, Mat. Zametki 37 (1985), no. 3, 391–398, 462 (Russian). MR 790428
B. V. Godun, Auerbach bases in Banach spaces that are isomorphic to $l_1[0,1]$, C. R. Acad. Bulgare Sci. 43 (1990), no. 10, 19–21 (1991) (Russian). MR 1106080
B. V. Godun, Bor-Luh Lin, and S. L. Troyanski, On Auerbach bases, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 115–118. MR 1209452, DOI 10.1090/conm/144/1209452
A. S. Granero, Almost-isometric copies of $(c_0(I), \| \cdot \|_\infty )$ or $(\ell _1(I), \|\cdot \|_1)$ in Banach spaces, unpublished note.
Antonio J. Guirao, Vicente Montesinos, and Václav Zizler, Open problems in the geometry and analysis of Banach spaces, Springer, [Cham], 2016. MR 3524558, DOI 10.1007/978-3-319-33572-8
Petr Hájek, Polynomials and injections of Banach spaces into superreflexive spaces, Arch. Math. (Basel) 63 (1994), no. 1, 39–44. MR 1277909, DOI 10.1007/BF01196297
Petr Hájek, Tomasz Kania, and Tommaso Russo, Symmetrically separated sequences in the unit sphere of a Banach space, J. Funct. Anal. 275 (2018), no. 11, 3148–3168. MR 3861732, DOI 10.1016/j.jfa.2018.01.008
Petr Hájek, Vicente Montesinos Santalucía, Jon Vanderwerff, and Václav Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 26, Springer, New York, 2008. MR 2359536
Petr Hájek and Matěj Novotný, Distortion of Lipschitz functions on $c_0(\Gamma )$, Proc. Amer. Math. Soc. 146 (2018), no. 5, 2173–2180. MR 3767367, DOI 10.1090/proc/13945
A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961/62), 123–128. MR 131986, DOI 10.4064/fm-50-2-123-128
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, Super-reflexive spaces with bases, Pacific J. Math. 41 (1972), 409–419. MR 308752
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
William B. Johnson, Joram Lindenstrauss, David Preiss, and Gideon Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3) 84 (2002), no. 3, 711–746. MR 1888429, DOI 10.1112/S0024611502013400
W. B. Johnson and H. P. Rosenthal, On $\omega ^{\ast }$-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. MR 310598, DOI 10.4064/sm-43-1-77-92
Jerzy Kąkol, Wiesław Kubiś, and Manuel López-Pellicer, Descriptive topology in selected topics of functional analysis, Developments in Mathematics, vol. 24, Springer, New York, 2011. MR 2953769, DOI 10.1007/978-1-4614-0529-0
Ondřej Kalenda, Stegall compact spaces which are not fragmentable, Topology Appl. 96 (1999), no. 2, 121–132. MR 1702306, DOI 10.1016/S0166-8641(98)00045-5
Ondřej Kalenda, Valdivia compacta and equivalent norms, Studia Math. 138 (2000), no. 2, 179–191. MR 1749079, DOI 10.4064/sm-138-2-179-191
Ondřej F. K. Kalenda, Valdivia compact spaces in topology and Banach space theory, Extracta Math. 15 (2000), no. 1, 1–85. MR 1792980
Ondřej F. K. Kalenda, A weak Asplund space whose dual is not in Stegall’s class, Proc. Amer. Math. Soc. 130 (2002), no. 7, 2139–2143. MR 1896051, DOI 10.1090/S0002-9939-02-06625-X
Tomasz Kania and Tomasz Kochanek, Uncountable sets of unit vectors that are separated by more than 1, Studia Math. 232 (2016), no. 1, 19–44. MR 3493287, DOI 10.4064/sm8353-2-2016
Petar S. Kenderov, Warren B. Moors, and Scott Sciffer, A weak Asplund space whose dual is not weak* fragmentable, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3741–3747. MR 1860511, DOI 10.1090/S0002-9939-01-06002-6
H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999), no. 2, 173–199. MR 1702641, DOI 10.1023/A:1009786603119
Clifford A. Kottman, Subsets of the unit ball that are separated by more than one, Studia Math. 53 (1975), no. 1, 15–27. MR 377477, DOI 10.4064/sm-53-1-15-27
Piotr Koszmider, Uncountable equilateral sets in Banach spaces of the form $C(K)$, Israel J. Math. 224 (2018), no. 1, 83–103. MR 3799750, DOI 10.1007/s11856-018-1637-9
Wiesław Kubiś, Banach spaces with projectional skeletons, J. Math. Anal. Appl. 350 (2009), no. 2, 758–776. MR 2474810, DOI 10.1016/j.jmaa.2008.07.006
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
A. J. Lazar, Points of support for closed convex sets, Illinois J. Math. 25 (1981), no. 2, 302–305. MR 607032
Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148. MR 160094, DOI 10.1007/BF02759700
Joram Lindenstrauss, On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967–970. MR 205040, DOI 10.1090/S0002-9904-1966-11606-3
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 97. Springer–Verlag, Berlin–New York, 1979.
S. K. Mercourakis and G. Vassiliadis, Equilateral sets in infinite dimensional Banach spaces, Proc. Amer. Math. Soc. 142 (2014), no. 1, 205–212. MR 3119196, DOI 10.1090/S0002-9939-2013-11746-6
Sophocles K. Mercourakis and Georgios Vassiliadis, Equilateral sets in Banach spaces of the form $C(K)$, Studia Math. 231 (2015), no. 3, 241–255. MR 3471052, DOI 10.4064/sm8259-1-2016
V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no. 6(162), 73–149 (Russian). MR 0420226
Warren B. Moors, Some more recent results concerning weak Asplund spaces, Abstr. Appl. Anal. 3 (2005), 307–318. MR 2197122, DOI 10.1155/AAA.2005.307
Warren B. Moors and Sivajah Somasundaram, Some recent results concerning weak Asplund spaces, Acta Univ. Carolin. Math. Phys. 43 (2002), no. 2, 67–86. 30th Winter School on Abstract Analysis (Lhota nade Rohanovem/Litice u České Lípy, 2002). MR 1979559
Warren B. Moors and Sivajah Somasundaram, A Gâteaux differentiability space that is not weak Asplund, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2745–2754. MR 2213755, DOI 10.1090/S0002-9939-06-08402-4
S. Negrepontis, Banach spaces and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1045–1142. MR 776642
A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 505–516. MR 438292, DOI 10.1112/jlms/s2-14.3.505
R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis 17 (1974), 78–90. MR 0352941, DOI 10.1016/0022-1236(74)90005-6
Robert R. Phelps, Convex functions, monotone operators and differentiability, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR 1238715
A. Pličko, On bounded biorthogonal systems in some function spaces, Studia Math. 84 (1986), no. 1, 25–37. MR 871844, DOI 10.4064/sm-84-1-25-37
S. Ruziewicz, Une généralisation d’un théorème de M. Sierpinski, Publ. Math. Univ. Belgrade 5 (1936), 23–27.
Saharon Shelah, Uncountable constructions for B.A., e.c. groups and Banach spaces, Israel J. Math. 51 (1985), no. 4, 273–297. MR 804487, DOI 10.1007/BF02764721
Charles Stegall, The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213–223. MR 374381, DOI 10.1090/S0002-9947-1975-0374381-1
Stevo Todorcevic, Biorthogonal systems and quotient spaces via Baire category methods, Math. Ann. 335 (2006), no. 3, 687–715. MR 2221128, DOI 10.1007/s00208-006-0762-7
S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180
J. Vanderwerff, J. H. M. Whitfield, and V. Zizler, Markuševič bases and Corson compacta in duality, Canad. J. Math. 46 (1994), no. 1, 200–211. MR 1260344, DOI 10.4153/CJM-1994-007-5
Neil H. Williams, Combinatorial set theory, Studies in Logic and the Foundations of Mathematics, vol. 91, North-Holland Publishing Co., Amsterdam, 1977. MR 3075383
V. Zizler, Locally uniformly rotund renorming and decompositions of Banach spaces, Bull. Austral. Math. Soc. 29 (1984), no. 2, 259–265. MR 744260, DOI 10.1017/S0004972700021493