Property (G) and (K) of Orlicz spaces. (English) Zbl 0724.46027

Wang, Tingfu

Commentat. Math. Univ. Carol. 31, No. 2, 307-313 (1990).

A Banach space X is said to have property (K) if the norm topology and the weak topology coincide on the unit sphere S(X). It is shown that Orlicz sequence space \(\ell_ M\) (resp. \(L_ M[0,1])\) equipped with Orlicz or Luxemburg norm have the property (K) iff M satisfies the (\(\Delta\) \({}_ 2)\) condition (resp. M satisfies the \((\Delta_ 2)\) condition and M is strictly convex on \([0,\infty)).\)
[Reviewer’s remark: In the abstract and in the beginning of the paper the author promises to prove that for Orlicz spaces the property (K) is equivalent to the following property (H): for any sequence on S(X), weak and norm convergence coincide. This seems to be wrong.]

Reviewer: V.Tarieladze (Tbilisi)

MSC:

46B25	Classical Banach spaces in the general theory
46B20	Geometry and structure of normed linear spaces
46B45	Banach sequence spaces

Keywords:

property (K); Orlicz sequence space; Orlicz or Luxemburg norm; \((\Delta _ 2)\) condition; strictly convex