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Smoothness and approximative compactness in Orlicz function spaces. (English) Zbl 1370.46013
Summary: Some criteria for approximative compactness of every weakly\(^*\) closed convex set of Orlicz function spaces equipped with the Luxemburg norm are given. Although, criteria for approximative compactness of Orlicz function spaces equipped with the Luxemburg norm were known, we can easily deduce them from our main results.

MSC:
46B50 Compactness in Banach (or normed) spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
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References:
[1] H. Hudzik and B.X. Wang, Approximative compactness in Orlicz spaces , J. Approx. Theory 95 (1998), no. 1, 82-89. · Zbl 0913.46032
[2] H. Hudzik, W. Kowalewski and G. Lewicki, Approximative compactness and full rotundity in Musielak-Orlicz space and Lorentz-Orlicz space , Z. Anal. Anwendungen 25 (2006), no. 1, 163-192. · Zbl 1108.46016
[3] S.T. Chen, Geometry of Orlicz spaces, Dissertations Math, Warszawa, 1996. · Zbl 1089.46500
[4] S. Tian and T. Wang, A characterization of extreme points in Orlicz function spaces, Journal of mathematical research and exposition, 16 (1996), no. 1, 81-89.(in Chinese) · Zbl 0848.46021
[5] S. Chen, H. Hudzik, W. Kowalewski, Y. Wang and M. Wisla, Approximative compactness and continuity of metric projector in Banach spaces and applications , Sci. China Ser A. 50 (2007), no. 2, 75-84. · Zbl 1153.46008
[6] N.W. Jefimow and S.B. Stechkin, Approximative compactness and Chebyshev sets , Sovit Math. 2 (1961), no. 2, 1226-1228.
[7] H. Hudzik and W. Kurc, Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices , J. Approx. Theory. 95 (1998), no. 3, 115-121. · Zbl 0921.41015
[8] S.T. Chen, X. He, H. Hudzik and A. Kamin’ska, Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm , J. Math. Anal. Appl. 344 (2008), no. 2, 687-698. · Zbl 1153.46014
[9] L. Jinlu, The generalized projection operator on reflexive Banach spaces and its applications , J. Math. Anal. Appl. 306 (2005), no. 1, 55-71. · Zbl 1129.47043
[10] S. Shang, Y. Cui and Y. Fu, Midpoint locally uniform rotundity of Musielak-Orlicz-Bochner function spaces endowed with the Luxemburg norm , J. Convex Anal. 19 (2012), no. 1, 213-223. · Zbl 1242.46019
[11] S. Shang, Y. Cui and Y. Fu, \(P\)-convexity of Orlicz-Bochner function spaces endowed with the Orlicz norm , Nonlinear Analysis. 74 (2012), no. 1, 371-379. · Zbl 1239.46015
[12] S. Shang and Y. Cui, Locally uniform convexity in Musielak-Orlicz function spaces of Bochner type endowed with the Luxemburg norm , J. Math. Anal. Appl. 378 (2011), no. 2, 432-441. · Zbl 1216.46015
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