Paul, Tanmoy
(2017)
Various notions of best approximation property in spaces of Bochner integrable functions.
Advances in Operator Theory, 2 (1).
pp. 59-77.
ISSN 2538-225X
Abstract
We show that a separable proximinal subspace of
X
, say
Y
is
strongly proximinal (strongly ball proximinal) if and only if
L
p
(
I, Y
) is strongly
proximinal (strongly ball proximinal) in
L
p
(
I, X
), for 1
≤
p <
∞
. The
p
=
∞
case requires a stronger assumption, that of ’uniform proximinality’. Further,
we show that
Y
is ball proximinal in
X
if and only if
L
p
(
I, Y
) is ball proximinal
in
L
p
(
I, X
) for 1
≤
p
≤ ∞
. We develop the notion of ’uniform proximinality’
of a closed convex set in a Banach space, rectifying one that was defined in
a recent paper by P.-K Lin et al. [J. Approx. Theory 183 (2014), 72–81].
We also provide several examples viz. any
U
-subspace of a Banach space
has this property. Recall the notion of 3
.
2
.I.P.
by Joram Lindenstrauss, a
Banach space
X
is said to have 3
.
2
.I.P.
if any three closed balls which are
pairwise intersecting actually intersect in
X
. It is proved the closed unit ball
B
X
of a space with 3
.
2
.I.P
and closed unit ball of any M-ideal of a space with
3
.
2
.I.P.
are uniformly proximinal. A new class of examples are given having
this property.
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