TY - JOUR

T1 - Intersection properties of the unit ball

AU - De Bernardi, Carlo Alberto

AU - Veselý, Libor

PY - 2019

Y1 - 2019

N2 - Let $X$ be a real Banach space with the closed unit ball $B_X$ and
the dual $X^{*}$. We say that $X$ has the {em intersection property}
$I$ ({em general intersection property} $GI$, respectively)
if, for each countable family (for each family, respectively)
${B_i}_{iin A}$ of equivalent closed unit balls such that
$B_X=igcap_{iin A} B_i$, one has $B_{X^{**}}=igcap_{iin A}
B_i^{circcirc}$, where $B_i^{circcirc}$ is the bipolar set of
$B_i$, that is, the bidual unit ball corresponding to $B_i$. In
this paper we study relations between properties $I$ and $GI$,
and geometric and differentiability properties of $X$.
For example, it follows by our results that if $X$ is Fr'echet
smooth or $X$ is a polyhedral Banach space then $X$ satisfies
property $GI$, and hence also property $I$. Moreover, for separable spaces $X$,
properties $I$ and $GI$ are equivalent and they imply that
$X$ has the ball generated property. However, properties $I$ and $GI$
are not equivalent in general. One of our main results
concerns $C(K)$ spaces: under certain topological condition on
$K$, satisfied for example by all zero-dimensional compact spaces
and hence by all scattered compact spaces, we prove that $C(K)$
satisfies $I$ if and only if every nonempty $G_delta$-subset of
$K$ has nonempty interior.

AB - Let $X$ be a real Banach space with the closed unit ball $B_X$ and
the dual $X^{*}$. We say that $X$ has the {em intersection property}
$I$ ({em general intersection property} $GI$, respectively)
if, for each countable family (for each family, respectively)
${B_i}_{iin A}$ of equivalent closed unit balls such that
$B_X=igcap_{iin A} B_i$, one has $B_{X^{**}}=igcap_{iin A}
B_i^{circcirc}$, where $B_i^{circcirc}$ is the bipolar set of
$B_i$, that is, the bidual unit ball corresponding to $B_i$. In
this paper we study relations between properties $I$ and $GI$,
and geometric and differentiability properties of $X$.
For example, it follows by our results that if $X$ is Fr'echet
smooth or $X$ is a polyhedral Banach space then $X$ satisfies
property $GI$, and hence also property $I$. Moreover, for separable spaces $X$,
properties $I$ and $GI$ are equivalent and they imply that
$X$ has the ball generated property. However, properties $I$ and $GI$
are not equivalent in general. One of our main results
concerns $C(K)$ spaces: under certain topological condition on
$K$, satisfied for example by all zero-dimensional compact spaces
and hence by all scattered compact spaces, we prove that $C(K)$
satisfies $I$ if and only if every nonempty $G_delta$-subset of
$K$ has nonempty interior.

KW - Geometry of Banach spaces

KW - Reflexivity

KW - Sequences of equivalent norms

KW - Space of continuous functions

KW - Geometry of Banach spaces

KW - Reflexivity

KW - Sequences of equivalent norms

KW - Space of continuous functions

UR - http://hdl.handle.net/10807/131780

U2 - https://doi.org/10.1016/j.jmaa.2019.03.008

DO - https://doi.org/10.1016/j.jmaa.2019.03.008

M3 - Article

SN - 0022-247X

SP - 1108

EP - 1129

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

ER -