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

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)
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