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Intersection properties of ball sequences and uniqueness of Hahn–Banach extensions

Published online by Cambridge University Press:  14 November 2011

E. Oja  and
M. Põldvere
E. Oja
Affiliation:
Faculty of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia (eveoja@math.ut.ee, mert@math.ut.ee)
M. Põldvere
Affiliation:
Faculty of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia (mert@math.ut.ee)
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Extract

Let X be a Banach space and Y a closed subspace. We introduce an intrinsic geometric property of Y—the k-ball sequence property—which is a weakening of the famous k-ball property due to Alfsen & Effros. We prove that Y satisfies the 2-ball sequence property if and only if Y has the Phelps uniqueness property U (i.e. every continuous linear functional g ∈Y* has a unique norm-preserving extension f ∈X*). We prove that Y is an ideal having property U if and only if Y satisfies the 3-ball sequence property, and in this case, Y satisfies the k-ball sequence property for all k.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1251 - 1262
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Alfsen, E. M. and Effros, E. G.. Structure in real Banach spaces. Ann. Math. 96 (1972), 98–173.CrossRefGoogle Scholar
2Beauzamy., B.. Introduction to Banach spaces and their geometry. North-Holland Mathematics Studies, vol. 68 (Amsterdam: North-Holland, 1982).Google Scholar
3Beauzamy, B. and Maurey., B.. Points minimaux et ensembles optimaux dans les espaces de Banach. J. Fund. Analysis 24 (1977), 107–139.CrossRefGoogle Scholar
4Behrends., E.. M-structure and the Banach-Stone theorem. Lecture Notes in Mathematics, vol. 736 (Springer, 1979).CrossRefGoogle Scholar
5Behrends., E.. Points of symmetry of convex sets in the two-dimensional complex space—a counterexample to D. Yost's problem. Math. Ann. 290 (1991), 463–471.Google Scholar
6Diestel., J.. Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics 485 (Springer, 1975).CrossRefGoogle Scholar
7Foguel, S. R.. On a theorem by A. E. Taylor. Proc. Am. Math. Soc. 9 (1958), 325.Google Scholar
8G. Godefroy, Kalton, N. J. and Saphar, P. D.. Unconditional ideals in Banach spaces. Studia Math. 104 (1993), 13–59.Google Scholar
9Harmand, P., Werner, D. and Werner., W.M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics 1547 (Springer, 1993).Google Scholar
10Hennefeld., J.M-ideals, HB-subspaces, and compact operators. Indiana Univ. Math. J. 28 (1979), 927–934.CrossRefGoogle Scholar
11Hennefeld., J.A note on M-ideals in B(X). Proc. Am. Math. Soc. 78 (1980), 89–92.Google Scholar
12Kakutani., S.Some characterzations of Euclidean space. Japan J. Math. 16 (1939), 93—97.Google Scholar
13Lima, Å.. Intersection properties of balls and subspaces in Banach spaces. Trans. Am. Math. Soc. 227 (1977), 1–62.CrossRefGoogle Scholar
14Lima, Å.. Uniqueness of Hahn—Banach extensions and liftings of linear dependences. Math. Scand. 53 (1983), 97–113.CrossRefGoogle Scholar
15Lindenstrauss, J. and Tzafriri., L.On the complemented subspaces problem. Israel J. Math. 9 (1971), 263–269.CrossRefGoogle Scholar
16Oja, E.. Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem. Mat. Zametki 43 (1988), 237–246 (in Russian). (English translation: Math. Notes 43 (1988), 134–139.)Google Scholar
17Oja., E.Dual de l'espace des opérateurs linéaires continus. C. R. Acad. Sci. Paris Sér. I 309 (1989), 983–986.Google Scholar
18Oja., E.H.B-subspaces and Godun sets of subspaces in Banach spaces. Mathematika 44 (1997), 120–132.Google Scholar
19Oja, E. and Põldvere, M.. On subspaces of Banach spaces where every functional has a unique norm-preserving extension. Studia Math. 117 (1996), 289–306.Google Scholar
20Phelps, R. R.. Uniqueness of Hahn—Banach extensions and unique best approximation. Trans. Am. Math. Soc. 95 (1960), 238–255.Google Scholar
21Sullivan., F.Geometrical properties determined by the higher duals of a Banach space. Illinois J. Math. 21 (1977), 315–331.CrossRefGoogle Scholar
22Taylor, A. E.. The extension of linear functionals. Duke Math. J. 5 (1939), 538–547.CrossRefGoogle Scholar
23Vlasov, L. P.. Approximative properties of sets in normed linear spaces. Russ. Math. Surv. 28 (1973), 3–66 (in Russian).Google Scholar
24Yost., D.The n-ball properties in real and complex Banach spaces. Math. Scand. 50 (1982), 100–110.Google Scholar