On CLUR points of Orlicz spaces

Abstract

CriteriaforcompactlylocallyuniformlyrotundpointsinOrliczspacesare given. Let (X, kk ) be a Banach space. S(X) and B(X) are the unit sphere and unit ball of X respectively. A point x on S(X) is said to be a locally uniformly rotund point (LUR point) provided that {xn} ⊂ X, k xnk → 1 and k xn + xk → 2 imply k xn − xk → 0. Recently Y. A. Cui, H. Hudzik and C. Meng (2) introduced the concept of compactly locally uniformly rotund point (CLUR point for short). x ∈ S(X) is said to be a CLUR point provided that {xn} ⊂ X, k xnk → 1 and k xn + xk → 2 imply that {xn} is a compact set. Obviously, if every point on S(X) is LUR (CLUR) then X is a LUR (CLUR) space. In 1975, B. B. Panda and O. P. Kapoor (4) proved that CLUR implies the Kadec-Klee property and X is LUR iff X is CLUR and strictly rotund. In 1984, J. Y. Fu and W. Y. Zhang (3) showed that if X ∗∗ is CLUR and K is a non-null closed, convex Chebyshev set in X then the projection on K is continuous. Y. A. Cui et al. (2) obtained a criterion for a point to be CLUR in Orlicz sequence spaces equipped with the Luxemburg norm. In this paper we will discuss the CLUR points in Orlicz sequence spaces equipped with the Orlicz norm and in Orlicz function spaces equipped with both the Luxemburg and Orlicz norm. A mapping M : (−∞, ∞) → (0, ∞) is said to be an N-function if it is even, convex, vanishing only at zero and such that limu→0 M(u)/u = 0 and limu→∞ M(u)/u = ∞. An interval (a, b) is called a structurally affine interval (SAI for short) of M provided that M is affine on ( a, b) and it is not affine on ( a − s, b) or (a, b + s) for any s > 0. For an N-function M we