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## Geometry of quotient spaces and proximinality

### Tom 82 / 2003

Annales Polonici Mathematici 82 (2003), 9-18 MSC: 46B20, 46E30, 46E40. DOI: 10.4064/ap82-1-2

#### Streszczenie

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 ${\mit \Phi }$ does not satisfy the $\delta _2$-condition, then $h_{{\mit \Phi }}^0$ is not proximinal in $l_{{\mit \Phi }}^0$ and the quotient space $l_{{\mit \Phi }}^0/ h_{{\mit \Phi }}^0$ is not rotund (even if $l_{{\mit \Phi }}^0$ 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.

#### Autorzy

• Yuan CuiDepartment of Mathematics
Harbin University of Sciences
and Technology
Harbin, P.R. China
• Henryk HudzikFaculty of Mathematics and Computer Science
Umultowska 87
61-614 Poznań, Poland
e-mail
• Yaowaluck KhongthamFaculty of Science
Maejo University
Chiang Mai, Thailand

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