DOI: 10.68381/jca22034 ISSN: 0944-6532
Ball Proximinal and Strongly Ball Proximinal Spaces
Pei-Kee Lin, Wen Zhang, Bentuo Zheng
Let
Y
Y
be an
E
E
-proximinal (respectively, a strongly proximinal) subspace of
X
X
. We prove that
Y
Y
is (strongly) ball proximinal in
X
X
if and only if for any
x\in X
x
∈
X
with
(x+Y)\cap B_X\ne\emptyset
(
x
+
Y
)
∩
B
X
≠
∅
,
(x+Y)\cap B_X
(
x
+
Y
)
∩
B
X
is (strongly) proximinal in
x+Y
x
+
Y
. Using this characterization and a smart construction, we obtain three Banach spaces
Z\subset Y\subset X
Z
⊂
Y
⊂
X
such that
Z
Z
is ball proximinal in
X
X
and
Y/Z
Y
/
Z
is ball proximinal in
X/Z
X
/
Z
, but
Y
Y
is not ball proximinal in
X
X
. This solves a problem raised by P. Bandyopadhyay, Bor-Luh Lin and T.S.S.R.K. Rao [Ball proximinality in Banach spaces, in: Banach Spaces and Their Applications in Analysis (Oxford/USA, 2006) B. Randrianantoanina et al (eds.) Proceedings in Mathematics, de Gruyter, Berlin (2007) 251–264].