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Property \((H)\) in Köthe-Bochner spaces. (English) Zbl 0871.46013

Summary: The property (H) in Köthe-Bochner space \(E(X)\), where \(E\) is a locally uniformly rotund Köthe function space and \(X\) is an arbitrary Banach space, is discussed. Specifically, the question of whether or not this geometrical property lifts from \(X\) to \(E(X)\) is examined. Among others it is proved that \(E(X)\) has the property (H) whenever \(X\) has the property (G). Moreover, it is shown that the property (H) does not lift from \(X\) to \(E(X)\) when the Köthe space \(E\) is over a measure space in which the measure is not purely atomic.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
46B20 Geometry and structure of normed linear spaces
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[1] Amrani, A.; Castaing, C.; Valadier, M., Méthodes de troncature appliquées à des problèmes de convergence faible ou forte dans \(L^1\), Arch. Rational Mech. Anal., 117, 167-191 (1992) · Zbl 0778.46016
[2] Benabdellah, H., Extrémalité et entaillabilité sur des convexes fermés non nécessairement bornés d’un espace de Banach. Caractérisations dans le cas des espaces intégraux, Sém. d’Anal. Convexe, Montpellier, exposé, 5 (1991) · Zbl 0825.46004
[3] Buhvalov, A. V., On some analytic representation of operators with abstract norm, Soviet. Math. Doklady, 14, 197-201 (1973) · Zbl 0283.47028
[4] Castaing, C. and R. Pluciennik — Property (H) in Köthe-Bochner sequence spaces. Submitted for publication.; Castaing, C. and R. Pluciennik — Property (H) in Köthe-Bochner sequence spaces. Submitted for publication. · Zbl 0871.46013
[5] Day, M. M., Some more uniformly convex spaces, Bull. Amer. Math. Soc., 47, 504-507 (1941) · JFM 67.0402.03
[6] Day, M. M., Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc., 78, 516-528 (1955) · Zbl 0068.09101
[7] Day, M. M., Normed linear spaces, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21 (1973), Springer-Verlag) · Zbl 0082.10603
[8] Fan, K.; Glicksberg, I., Some geometric properties of the spheres in a normed linear space, Duke Math. J., 25, 553-568 (1958) · Zbl 0084.33101
[9] Halperin, I., Uniform convexity in function spaces, Duke Math. J., 21, 195-204 (1954) · Zbl 0055.33702
[10] Hiai, F., Representation of additive functionals on vector-valued normed Köthe spaces, Kodai Math. J., 2, 3, 300-313 (1979) · Zbl 0431.46025
[11] Hudzik, H.; Landes, T., Characteristic of convexity of Köthe function spaces, Math. Ann., 294, 117-124 (1992) · Zbl 0761.46016
[12] Hudzik, H.; Mastylo, M., Strongly extreme points in Köthe-Bochner spaces, Rocky Mtn. J. Math., 23, 3, 899-909 (1993) · Zbl 0795.46017
[13] Huff, R., Banach spaces which are nearly uniformly convex, Rocky Mtn J. Math., 10, 743-749 (1980) · Zbl 0505.46011
[14] Kadec, M. I., Relations between some properties of convexity of the unit ball of a Banach space, Functional Anal. and Appl., 16, 204-206 (1982)
[15] Kamińska, A., Some convexity properties of Musielak-Orlicz spaces of Bochner type Suplemento ai Rendiconti del Circolo Matematica di Palermo, The 13-th Winter School in Srni 1985, 10, 63-73 (1985), Serie II · Zbl 0609.46015
[16] Kamińska, A.; Turett, B., Rotundity in Köthe spaces of vector-valued functions, Can. J. Math., 41, 4, 659-675 (1989) · Zbl 0668.46013
[17] Kantorovic, L. V.; Akilov, G. P., Functional Analysis (in Russian) (1978), Moscow
[18] Leonard, I. E., Banach sequence spaces, J. Math. Anal. Appl., 54, 245-265 (1976) · Zbl 0343.46010
[19] Lin, B. L.; Lin, P. K., Denting points in Bochner \(L^p\) spaces, (Proc. Amer. Math. Soc., 97 (1986)), 629-633, (4) · Zbl 0603.46041
[20] Lin, B. L.; Lin, P. K., Property (H) in Lebesgue-Bochner function spaces, (Proc. Amer. Math. Soc., 95 (1985)), 581-584, (4) · Zbl 0595.46021
[21] Lin, B. L.; Lin, P. K.; Troyanski, S. L., Characterization of denting points, (Proc. Amer. Math. Soc., 102 (1988)), 526-528, (3) · Zbl 0649.46015
[22] Lin, B. L.; Lin, P. K.; Troyanski, S. L., Some geometric and topological properties of the unit sphere in a Banach space, Math. Ann., 274, 613-616 (1986) · Zbl 0770.46004
[23] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces II (1979), Springer-Verlag · Zbl 0403.46022
[24] Lovalgia, A. R., Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc., 78, 225-238 (1955) · Zbl 0064.35601
[25] Mazur, S., Über konvexe Mengen in linearen normierte Räumen, Studia Math., 4, 70-84 (1933) · Zbl 0008.31603
[26] McShane, E. J., Linear functionals on certain Banach spaces, (Proc. Amer. Math. Soc., 1 (1950)), 402-408 · Zbl 0039.11802
[27] Pluciennik, R.; Wang, T. F.; Zhang, Y. L., \(H\)-points and denting points in Orlicz spaces, Comment. Math., 33, 135-151 (1993) · Zbl 0793.46012
[28] Radon, J., Theorie und Anvendungen der absolut additiven Mengen funktionen, Sitzungsber. Akad. Wiss. Wien, 122, 1295-1438 (1913) · JFM 44.0464.03
[29] Riesz, F., Sur la convergence en moyenne II, Acta Sci. Math. (Szeged), 4, 182-185 (1928-1929) · JFM 55.0156.01
[30] Smith, M. A., Strongly extreme points in \(L^p(μ, X)\), Rocky Mtn. J. Math., 16, 1-5 (1986) · Zbl 0601.46021
[31] Smith, M. A., Rotundity and extremity in \(l^p(X)\) and \(L^p(μ,X)\), Contemporary Math., 52, 143-162 (1986)
[32] Smith, M. A.; Turett, B., Rotundity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc., 257, 105-118 (1980) · Zbl 0368.46039
[33] Turett, B., Rotundity of Orlicz spaces, Indagationes Math., 38, 462-469 (1976) · Zbl 0388.46022
[34] Valadier, M., Différents cas où, grâce à une propriété d’extrémalité, une suite de fonctions intégrables faiblement convergente converge fortement, Sém. d’Anal. Convexe, Montpellier, exposé, 5 (1989) · Zbl 0734.49004
[35] Visintin, A., Strong convergence results related to strict convexity, Comm. Partial Differential Equations, 9, 439-466 (1984) · Zbl 0545.49019
[36] Zizler, V., On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.), 87 (1971) · Zbl 0231.46036
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