## Banach Journal of Mathematical Analysis

### Smoothness and approximative compactness in Orlicz function spaces

#### Abstract

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.

#### Article information

Source
Banach J. Math. Anal., Volume 8, Number 1 (2014), 26-38.

Dates
First available in Project Euclid: 14 October 2013

https://projecteuclid.org/euclid.bjma/1381782084

Digital Object Identifier
doi:10.15352/bjma/1381782084

Mathematical Reviews number (MathSciNet)
MR3161679

Zentralblatt MATH identifier
1370.46013

#### Citation

Shang, Shaoqiang; Cui, Yunan; Fu, Yongqiang. Smoothness and approximative compactness in Orlicz function spaces. Banach J. Math. Anal. 8 (2014), no. 1, 26--38. doi:10.15352/bjma/1381782084. https://projecteuclid.org/euclid.bjma/1381782084

#### References

• H. Hudzik and B.X. Wang, Approximative compactness in Orlicz spaces, J. Approx. Theory 95 (1998), no. 1, 82–89.
• 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.
• S.T. Chen, Geometry of Orlicz spaces, Dissertations Math, Warszawa, 1996.
• 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)
• 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.
• N.W. Jefimow and S.B. Stechkin, Approximative compactness and Chebyshev sets, Sovit Math. 2 (1961), no. 2, 1226–1228.
• 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.
• 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.
• L. Jinlu, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl. 306 (2005), no. 1, 55–71.
• 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.
• 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.
• 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.