×

The distance \(dist (\mathcal{B},X)\) when \(\mathcal{B}\) is a boundary of \(B(X^{**})\). (English) Zbl 1225.46011

Let \(X\) be a Banach space and let \(\mathcal{B}\) be a boundary of its double dual closed unit ball \(B_{X^{\ast\ast}}\). This means that \(\mathcal{B}\) is a subset of \(B_{X^{\ast\ast}}\) such that for every \(x^\ast\in X^{\ast}\) there exists \(b\in\mathcal{B}\) with \(b(x^\ast) = \|x^\ast\|\). For two sets \(A,C\), let \( \text{dist}(A,C) = \sup\{\inf\{\|a - c\| : a\in A\} : c\in C\}\}\). The main result states that, under these assumptions,
\[ \text{dist}(\mathcal{B},X) = \text{dist}(B_{X^{\ast\ast}},X). \]
Notice that both distances are just zero when \(X\) is reflexive, and the distance in the right equals 1 when \(X\) is not reflexive, so the main point is that \(\text{dist}(\mathcal{B},X) = 1\) when \(X\) is not reflexive. This result is related to previous research on the relation of \(\text{dist}(K,X)\) and \(\text{dist}(\overline{co}^{w^\ast}(K),X)\) when \(K\subset X^{\ast\ast}\) is a weak\(^\ast\) compact set, as well as on the relation between \(\mathcal{B}\) and \(\overline{co}^{w^\ast}(\mathcal{B})\) for a general boundary \(\mathcal{B}\).

MSC:

46B26 Nonseparable Banach spaces
46B20 Geometry and structure of normed linear spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. · Zbl 0542.46007
[2] M. Fabian, P. Hájek, V. Montesinos, and V. Zizler, A quantitative version of Krein’s theorem, Rev. Mat. Iberoamericana 21 (2005), no. 1, 237 – 248. · Zbl 1083.46012 · doi:10.4171/RMI/421
[3] V. P. Fonf, J. Lindenstrauss, and R. R. Phelps, Infinite dimensional convexity, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 599 – 670. · Zbl 1086.46004 · doi:10.1016/S1874-5849(01)80017-6
[4] Gilles Godefroy, Boundaries of a convex set and interpolation sets, Math. Ann. 277 (1987), no. 2, 173 – 184. · Zbl 0597.46015 · doi:10.1007/BF01457357
[5] Antonio S. Granero, An extension of the Krein-Šmulian theorem, Rev. Mat. Iberoam. 22 (2006), no. 1, 93 – 110. · Zbl 1117.46002 · doi:10.4171/RMI/450
[6] A. S. Granero, P. Hájek, and V. Montesinos Santalucía, Convexity and \?*-compactness in Banach spaces, Math. Ann. 328 (2004), no. 4, 625 – 631. · Zbl 1059.46015 · doi:10.1007/s00208-003-0496-8
[7] William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 1 – 84. · Zbl 1011.46009 · doi:10.1016/S1874-5849(01)80003-6
[8] Marianne Morillon, A new proof of James’ sup theorem, Extracta Math. 20 (2005), no. 3, 261 – 271. · Zbl 1121.46013
[9] Stephen Simons, An eigenvector proof of Fatou’s lemma for continuous functions, Math. Intelligencer 17 (1995), no. 3, 67 – 70. · Zbl 0841.26006 · doi:10.1007/BF03024373
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.