On some geometric properties of certain Köthe sequence spaces

Cui, Yunan, Hudzik, Henryk, and Zhang, Tao. "On some geometric properties of certain Köthe sequence spaces." Mathematica Bohemica 124.2-3 (1999): 303-314. <http://eudml.org/doc/248460>.

@article{Cui1999,
abstract = {It is proved that if a Kothe sequence space $X$ is monotone complete and has the weakly convergent sequence coefficient WCS$(X)>1$, then $X$ is order continuous. It is shown that a weakly sequentially complete Kothe sequence space $X$ is compactly locally uniformly rotund if and only if the norm in $X$ is equi-absolutely continuous. The dual of the product space $(\bigoplus \nolimits _\{i=1\}^\{\infty \}X_i)_\{\Phi \}$ of a sequence of Banach spaces $(X_i)_\{i=1\}^\{\infty \}$, which is built by using an Orlicz function $\Phi $ satisfying the $\Delta _2$-condition, is computed isometrically (i.e. the exact norm in the dual is calculated). It is also shown that for any Orlicz function $\Phi $ and any finite system $X_1,\dots ,X_n$ of Banach spaces, we have $\mathop WCS((\bigoplus \nolimits _\{i=1\}^nX_i)_\{\Phi \})=\min \lbrace \mathop WCS(X_i) i=1,\dots ,n\rbrace $ and that if $\Phi $ does not satisfy the $\Delta _2$-condition, then WCS$((\bigoplus \nolimits _\{i=1\}^\{\infty \}X_i) _\{\Phi \})=1$ for any infinite sequence $(X_i)$ of Banach spaces.},
author = {Cui, Yunan, Hudzik, Henryk, Zhang, Tao},
journal = {Mathematica Bohemica},
keywords = {Köthe sequence space; weakly convergent sequence coefficient; order continuity of the norm; absolute continuity of the norm; compact local uniform rotundity; Orlicz sequence space; Luxemburg norm; Orlicz norm; dual space; product space; Köthe sequence space; weakly convergent sequence coefficient; order continuity of the norm; absolute continuity of the norm; compact local uniform rotundity; Orlicz sequence space; Luxemburg norm; Orlicz norm; dual space; product space},
language = {eng},
number = {2-3},
pages = {303-314},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some geometric properties of certain Köthe sequence spaces},
url = {http://eudml.org/doc/248460},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Cui, Yunan
AU - Hudzik, Henryk
AU - Zhang, Tao
TI - On some geometric properties of certain Köthe sequence spaces
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 303
EP - 314
AB - It is proved that if a Kothe sequence space $X$ is monotone complete and has the weakly convergent sequence coefficient WCS$(X)>1$, then $X$ is order continuous. It is shown that a weakly sequentially complete Kothe sequence space $X$ is compactly locally uniformly rotund if and only if the norm in $X$ is equi-absolutely continuous. The dual of the product space $(\bigoplus \nolimits _{i=1}^{\infty }X_i)_{\Phi }$ of a sequence of Banach spaces $(X_i)_{i=1}^{\infty }$, which is built by using an Orlicz function $\Phi $ satisfying the $\Delta _2$-condition, is computed isometrically (i.e. the exact norm in the dual is calculated). It is also shown that for any Orlicz function $\Phi $ and any finite system $X_1,\dots ,X_n$ of Banach spaces, we have $\mathop WCS((\bigoplus \nolimits _{i=1}^nX_i)_{\Phi })=\min \lbrace \mathop WCS(X_i) i=1,\dots ,n\rbrace $ and that if $\Phi $ does not satisfy the $\Delta _2$-condition, then WCS$((\bigoplus \nolimits _{i=1}^{\infty }X_i) _{\Phi })=1$ for any infinite sequence $(X_i)$ of Banach spaces.
LA - eng
KW - Köthe sequence space; weakly convergent sequence coefficient; order continuity of the norm; absolute continuity of the norm; compact local uniform rotundity; Orlicz sequence space; Luxemburg norm; Orlicz norm; dual space; product space; Köthe sequence space; weakly convergent sequence coefficient; order continuity of the norm; absolute continuity of the norm; compact local uniform rotundity; Orlicz sequence space; Luxemburg norm; Orlicz norm; dual space; product space
UR - http://eudml.org/doc/248460
ER -