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2003 | 82 | 1 | 9-18
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Geometry of quotient spaces and proximinality

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It is proved that if X is a rotund Banach space and M is a closed and proximinal subspace of X, then the quotient space X/M is also rotund. It is also shown that if Φ does not satisfy the δ₂-condition, then $h⁰_{Φ}$ is not proximinal in $l⁰_{Φ}$ and the quotient space $l⁰_{Φ}/h⁰_{Φ}$ is not rotund (even if $l⁰_{Φ}$ is rotund). Weakly nearly uniform convexity and weakly uniform Kadec-Klee property are introduced and it is proved that a Banach space X is weakly nearly uniformly convex if and only if it is reflexive and it has the weakly uniform Kadec-Klee property. It is noted that the quotient space X/M with X and M as above is weakly nearly uniformly convex whenever X is weakly nearly uniformly convex. Criteria for weakly nearly uniform convexity of Orlicz sequence spaces equipped with the Orlicz norm are given.
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  • Department of Mathematics, Harbin University of Sciences, and Technology, Harbin, P.R. China
  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
  • Faculty of Science, Maejo University, Chiang Mai, Thailand
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bwmeta1.element.bwnjournal-article-doi-10_4064-ap82-1-2
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