Some remarks on stronger versions of the Boundary Problem for Banach spaces
Keywords:Boundary, Weak compactness, Convex hull, Extreme points, e- weakly relatively compact sets, e-interchangeable double limits
Let X be a real Banach space. A subset B of the dual unit sphere of X is said to be a boundary for X, if every element of X attains its norm on some functional in B. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of X which is compact in the topology of pointwise convergence on B is already weakly compact. This problem was recently solved by Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.
F. Albiac and N. Kalton, Topics in Banach Space Theory, Springer Graduate Texts in Mathematics vol.233, 2006
C. Angosto and B. Cascales, Measures of weak noncompactness in Banach spaces, Topology Appl. 156, no. 7 (2009), 1412–1421. http://dx.doi.org/10.1016/j.topol.2008.12.011
E. Behrends, New proofs of Rosenthal’s â„“1-theorem and the Josefson-Nissenzweig theorem, Bull. Polish Acad. Sci. Math. 43 (1996), 283–295.
J. Bourgain and M. Talagrand, Compacité extremalé, Proc. Amer. Math. Soc. 80 (1980), 68–70. http://dx.doi.org/10.1090/S0002-9939-1980-0574510-8
B. Cascales and G. Godefroy, Angelicity and the boundary problem, Mathematika 45 (1998), 105–112. http://dx.doi.org/10.1112/S0025579300014066
B. Cascales, O. Kalenda and J. Spurný, A quantitative version of James’ compactness theorem, (http://arxiv.org/abs/1005.5693).
B. Cascales, W. Marciszewski and M. Raja, Distance to spaces of continuous functions, Topology Appl. 153, no. 13 (2006), 2303–2319. http://dx.doi.org/10.1016/j.topol.2005.07.002
B. Cascales, M. Muñoz and J. Orihuela, James boundaries and σ-fragmented selectors, Studia Math. 188, no. 2 (2008), 97–122. p://dx.doi.org/10.4064/sm188-2-1
B. Cascales and R. Shvydkoy, On the Krein-Smulian Theorem for weaker topologies, Illinois J. Math. 47 (2003), 957–976.
N. Dunford and J. Schwartz, Linear Operators Part I: General Theory, Interscience Publishers, New York, 1958
M. Fabian, P. Hájek, V. Montesinos and V. Zizler, A quantitative version of Krein’s theorem, Rev. Mat. Iberoamericana 21, no. 1 (2005), 237–248. http://dx.doi.org/10.4171/RMI/421
K. Floret, Weakly compact sets, Springer Lectures Notes in Mathematics vol. 801, 1980
V. P. Fonf, J. Lindenstrauss and R. R. Phelps, Infinite dimensional convexity, in: Handbook of the geometry of Banach spaces, vol. 1 (W. B. Johnson and J. Lindenstrauss, eds.), North-Holland, 2001, pp. 599–670. http://dx.doi.org/10.1016/S1874-5849(01)80017-6
G. Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann. 277 (1987), 173–184. http://dx.doi.org/10.1007/BF01457357
G. Godefroy, Five lectures in Geometry of Banach spaces, Seminar on Functional Analysis (1987), 9–67.
R. B. Holmes, Geometric Functional Analysis and its Applications, Springer Graduate Texts in Mathematics vol. 24, 1975. http://dx.doi.org/10.1007/978-1-4684-9369-6
O. F. K. Kalenda, H. Pfitzner and J. Spurný, On quantification of weak sequential completeness, preprint, 2010, (http://www.arxiv.org/abs/1011.6553v1).
W. B. Moors, A characterisation of weak compactness in Banach spaces, Bull. Austral. Math. Soc. 55 (1997), 497–501. http://dx.doi.org/10.1017/S0004972700034146
W. B. Moors and J. Spurný, On the topology of pointwise convergence on the boundaries of L1-preduals, Proc. Amer. Math. Soc. 137 (2009), 1421–1429. http://dx.doi.org/10.1090/S0002-9939-08-09708-6
H. Pfitzner, Boundaries for Banach spaces determine weak compactness, Invent. Math. 182 (2010), 585–604. http://dx.doi.org/10.1007/s00222-010-0267-6
J. Rainwater, Weak convergence of bounded sequences, Proc. Amer. Math. Soc. 14 (1963), 999.
S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703–708. http://dx.doi.org/10.2140/pjm.1972.40.703
S. Simons, An eigenvector proof of Fatou’s lemma for continuous functions, Math. Intelligencer 17 (1995), 67–70. http://dx.doi.org/10.1007/BF03024373
J. Spurný, The Boundary Problem for L1-Preduals, Illinois J. Math. 52 (2008), 1183–1193.
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