Banach spaces which are isometric to subspaces of \(c_0(\Gamma)\). (English) Zbl 1482.46007
Characterizing those Banach spaces which can be found as subspaces of \(c_0\) is a long-standing problem in the mathematical literature; see, for instance, [S. A. Argyros et al., Mathematika 62, No. 3, 685–700 (2016; Zbl 1359.46005); G. Godefroy et al., Geom. Funct. Anal. 10, No. 4, 798–820 (2000; Zbl 0974.46023)]. In this article the authors focus on the isometric characterization of these spaces in terms of their extremal structure. One of the main results of the paper is the following:
{Theorem.} Suppose that \(X\) is a separable Banach space. Then the following statements are equivalent:
Notice that some of the equivalences of the previous theorem might belong to the folklore; see, e.g., Proposition III.1 in [G. Godefroy, Extr. Math. 16, No. 1, 1–25 (2001; Zbl 0986.46009)]. The authors also prove several properties of subspaces of \(c_0\) and deal with the nonseparable setting, i.e., they also study subspaces of \(c_0(\Gamma)\) for some uncountable set \(\Gamma\), providing several characterizations of such spaces.
{Theorem.} Suppose that \(X\) is a separable Banach space. Then the following statements are equivalent:
- 1.
- \(X\) is isometric to a subspace of \(c_0\).
- 2.
- The set of all extreme points of \(B_{X^*}\) is a weak*-null sequence in \(X^*\).
- 3.
- The set of all Gateâux derivatives of the norm \(\|\cdot \|\) of \(X\) is a weak*-null sequence.
- 4.
- The set of all Fréchet derivatives of the norm \(\|\cdot \|\) of \(X\) is a weak*-null sequence and a boundary for \(X\).
- 5.
- There is a weak*-null \(1\)-norming sequence in \(X^*\).
Notice that some of the equivalences of the previous theorem might belong to the folklore; see, e.g., Proposition III.1 in [G. Godefroy, Extr. Math. 16, No. 1, 1–25 (2001; Zbl 0986.46009)]. The authors also prove several properties of subspaces of \(c_0\) and deal with the nonseparable setting, i.e., they also study subspaces of \(c_0(\Gamma)\) for some uncountable set \(\Gamma\), providing several characterizations of such spaces.
Reviewer: Gonzalo Martínez Cervantes (Cartagena)
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B25 | Classical Banach spaces in the general theory |
| 46B26 | Nonseparable Banach spaces |
Keywords:
isometric embedding to \(c_0\); Fréchet derivative; Gâteaux derivative; extreme point; Banach spaceReferences:
| [1] | Aharoni, I., Every separable metric space is Lipschitz equivalent to a subset of c_0, Israel J. Math., 19, 284-291 (1974) · Zbl 0303.46012 · doi:10.1007/BF02757727 |
| [2] | Amir, D.; Lindenstrauss, J., The structure of weakly compact sets in Banach spaces, Ann. of Math., 88, 2, 35-46 (1968) · Zbl 0164.14903 · doi:10.2307/1970554 |
| [3] | Assouad, P., Remarques sur un article de Israel Aharoni sur les prolongements lipschitziens dans c_0. (Israel J. Math., 19, 284-291 (1974)), Israel J. Math., 31, 1, 97-100 (1978) · Zbl 0387.54003 · doi:10.1007/BF02761384 |
| [4] | Baudier, F.; Lancien, G.; Schlumprecht, Th, The coarse geometry of Tsirelson’s space and applications, J. Amer. Math. Soc., 31, 3, 699-717 (2018) · Zbl 1396.46020 · doi:10.1090/jams/899 |
| [5] | Chen, X.; Wang, Q.; Yu, G., The coarse Novikov conjecture and Banach spaces with Property (H), J. Funct. Anal., 268, 9, 2754-2786 (2015) · Zbl 1335.46060 · doi:10.1016/j.jfa.2015.02.001 |
| [6] | Cheng, L.; Cheng, Q.; Luo, S., On super weak compactness of subsets and its equivalences in Banach spaces, J. Convex Anal., 25, 3, 899-926 (2018) · Zbl 1408.46014 |
| [7] | Corson, H. H.; Lindenstrauss, J., On weakly compact subsets of Banach spaces, Proc. Amer. Math. Soc., 17, 407-412 (1966) · Zbl 0186.44703 · doi:10.1090/S0002-9939-1966-0199669-9 |
| [8] | Dowling, P. N.; Turentt, B., Lindenstrauss-Phelps spaces and the optimality of a theorem of Fonf, J. Nonlinear Convex Anal., 17, 2339-2342 (2016) · Zbl 1366.46009 |
| [9] | Dutrieux, Y.; Lancien, G., Isometric embeddings of compact spaces into Banach spaces, J. Funct. Anal., 255, 2, 494-501 (2008) · Zbl 1163.46004 · doi:10.1016/j.jfa.2008.04.002 |
| [10] | Fabian, M., Gâteaux Differentiability of Convex Functions and Topology, Weak Asplund Spaces (1997), New York: A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York · Zbl 0883.46011 |
| [11] | Fabian, M.; González, A.; Zizler, V., Flat sets, ℓ_p-generating and fixing c_0 in the nonseparable setting, J. Aust. Math. Soc., 87, 2, 197-210 (2009) · Zbl 1194.46030 · doi:10.1017/S1446788709000068 |
| [12] | Fonf, V. P., A property of Lindenstrauss-Phelps spaces, Funktsional. Anal. i Prilozhen., 13, 1, 79-80 (1979) · Zbl 0426.46012 · doi:10.1007/BF01076450 |
| [13] | Fonf, V. P., Some properties of polyhedral Banach spaces, Funktsional. Anal. i Prilozhen., 14, 4, 89-90 (1980) · Zbl 0454.46016 |
| [14] | Godefroy, G.; Kalton, N. J.; Lancien, G., On subspaces of c_0 and extensions of operators into C(K)-spaces, Quart. J. Math (Oxford), 52, 313-328 (2001) · Zbl 1016.46012 · doi:10.1093/qjmath/52.3.313 |
| [15] | Kalton, N. J.; Lancien, G., Best constants for Lipschitz embeddings of metric spaces into c_0, Fund. Math., 199, 3, 249-272 (2008) · Zbl 1153.46047 · doi:10.4064/fm199-3-4 |
| [16] | Lindentrauss, J.; Phelps, R. R., Extreme point properties of convex bodies in reflexive Banach spaces, Isreal J. Math., 6, 1, 39-48 (1968) · Zbl 0157.43802 · doi:10.1007/BF02771604 |
| [17] | Lindentrauss, J.; Preiss, D., On Fréchet differentiability of Lipschitz maps between Banach spaces, Ann. of Math., 157, 1, 257-288 (2003) · Zbl 1171.46313 · doi:10.4007/annals.2003.157.257 |
| [18] | Martìn, M., On proximinality of subspaces and the lineability of the set of norm-attaining functionals of Banach spaces, J. Funct. Anal., 278, 4, 108353 (2020) · Zbl 1435.46008 · doi:10.1016/j.jfa.2019.108353 |
| [19] | Nathansky, H. F.; Medina, F. N.; Bedoya, J. A V., Some comments about the paper “Lindestrauss-Phelps spaces and the optimality of a theorem of Fonf”, J. Nonlinear Convex Anal., 20, 103-109 (2019) · Zbl 1468.46023 |
| [20] | Pelant, X., Embeddings into c_0, Topology Appl., 57, 2-3, 259-269 (1994) · Zbl 0865.54025 · doi:10.1016/0166-8641(94)90053-1 |
| [21] | Phelps, R. R., Convex Functions, Monotone Operators and Differentiability (1989), Berlin: Springer-Verlag, Berlin · Zbl 0658.46035 · doi:10.1007/978-3-662-21569-2 |
| [22] | Rainwater, J., Local uniform convexity of Day’s norm on c_0(Γ), Proc. Amer. Math. Soc., 22, 335-339 (1969) · Zbl 0185.37602 |
| [23] | Read, C. J., Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math., 223, 1, 493-504 (2018) · Zbl 1397.46010 · doi:10.1007/s11856-017-1627-3 |
| [24] | Rmoutil, M., Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal., 272, 3, 918-928 (2017) · Zbl 1359.46008 · doi:10.1016/j.jfa.2016.10.028 |
| [25] | Swift, A., On coarse Lipschitz embeddability into c_0(k), Fund. Math., 241, 1, 67-81 (2018) · Zbl 1409.46015 · doi:10.4064/fm383-3-2017 |
| [26] | Tsirelson, B. S., It is impossible to imbed ℓ_p or c_0 into an arbitrary Banach space, Funktsional. Anal. i Prilozhen., 8, 2, 57-60 (1974) |
| [27] | Wu, C. X.; Cheng, L. X.; Yao, X. B., Characterizations of differentiability points of norms on c_0(Γ) and ℓ_∞(Γ), Northeast. Math. J., 12, 2, 153-160 (1996) · Zbl 0861.46012 |
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