Geometry of quotient spaces and proximinality

Yuan Cui ; Henryk Hudzik ; Yaowaluck Khongtham

Annales Polonici Mathematici, Tome 81 (2003), p. 9-18 / Harvested from The Polish Digital Mathematics Library

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 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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Publié le : 2003-01-01

Zbl 1107.46010

EUDML-ID : urn:eudml:doc:280733