Geometry of quotient spaces and proximinality

Yuan Cui, Henryk Hudzik, and Yaowaluck Khongtham. "Geometry of quotient spaces and proximinality." Annales Polonici Mathematici 82.1 (2003): 9-18. <http://eudml.org/doc/280733>.

@article{YuanCui2003,
abstract = {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.},
author = {Yuan Cui, Henryk Hudzik, Yaowaluck Khongtham},
journal = {Annales Polonici Mathematici},
keywords = {quotient space; proximinality; Orlicz sequence space; Orlicz norm; Köthe sequence space; rotundity; weakly uniform Kadec-Klee property; weakly nearly uniform convexity; nearly uniform convexity},
language = {eng},
number = {1},
pages = {9-18},
title = {Geometry of quotient spaces and proximinality},
url = {http://eudml.org/doc/280733},
volume = {82},
year = {2003},
}

TY - JOUR
AU - Yuan Cui
AU - Henryk Hudzik
AU - Yaowaluck Khongtham
TI - Geometry of quotient spaces and proximinality
JO - Annales Polonici Mathematici
PY - 2003
VL - 82
IS - 1
SP - 9
EP - 18
AB - 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.
LA - eng
KW - quotient space; proximinality; Orlicz sequence space; Orlicz norm; Köthe sequence space; rotundity; weakly uniform Kadec-Klee property; weakly nearly uniform convexity; nearly uniform convexity
UR - http://eudml.org/doc/280733
ER -