Geometry of quotient spaces and proximinality

Yuan Cui¹ ; Henryk Hudzik² ; Yaowaluck Khongtham³

¹ Department of Mathematics Harbin University of Sciences and Technology Harbin, P.R. China
² Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland
³ Faculty of Science Maejo University Chiang Mai, Thailand

Annales Polonici Mathematici, Tome 82 (2003) no. 1, pp. 9-18

Résumé

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.

Détail

DOI : 10.4064/ap82-1-2

Keywords: proved rotund banach space closed proximinal subspace quotient space rotund shown mit phi does satisfy delta condition mit phi proximinal mit phi quotient space mit phi mit phi rotund even mit phi rotund weakly nearly uniform convexity weakly uniform kadec klee property introduced proved banach space weakly nearly uniformly convex only reflexive has weakly uniform kadec klee property noted quotient space above weakly nearly uniformly convex whenever weakly nearly uniformly convex criteria weakly nearly uniform convexity orlicz sequence spaces equipped orlicz norm given