The coordinatewise uniformly Kadec-Klee property in some Banach spaces. (Russian, English) Zbl 1033.46027

Zhang, Tao

Sib. Mat. Zh. 44, No. 2, 454-458 (2003); translation in Sib. Math. J. 44, No. 2, 363-365 (2003).

Summary: We introduce a new property \(UKK_c\) for a Banach space and show that the following three properties are equivalent for an Orlicz sequence space: \(UKK_c\), \(H_c\), and \(\Phi\in\delta_2\). Besides, we prove that the direct Orlicz sums \(\bigl(\sum_{n=1}^{\infty}\oplus X_n\bigr)_{l_\Phi}\) and \(\bigl(\sum_{n=1}^{\infty}\oplus X_n\bigr)_{l_{(\Phi)}}\) possess the property \(H_c\) provided that each \(X_n\), \(n\in\mathbb N\), possesses the property \(H_c\) and \(\Phi\in\delta_2\).

MSC:

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
46A45	Sequence spaces (including Köthe sequence spaces)

Keywords:

Orlicz sequence space; Orlicz-direct-sum space; geometric properties of Köthe sequence space