Proximinality in τ -ALUR Banach spaces Definition 1.
P. Bandyopadhyay et al. J. Math. Anal. Appl. 341 2008 309–317 315
3. Proximinality in τ -ALUR Banach spaces Definition 3.1.
See [2]. Recall that a Banach space X is τ -almost locally uniformly rotund τ -ALUR if for any x ∈ S
X
, {x
n
} ⊆ B
X
and {x
∗ m
} ⊆ B
X
∗
, the condition lim
m
lim
n
x
∗ m
x
n
+ x 2
= 1
implies τ -lim
n
x
n
= x .
In the literature, the acronym ALUR is also used for average locally uniformly rotund which is known to be equivalent to X is strictly convex and has the KK property. We will use it in the above sense only. Theorem 3.4 below
discusses the relation between these two notions. In this section, we consider proximinality in τ -ALUR spaces.
From the results of [1, Proposition 4.4 and Corollary 4.6], it follows that a Banach space X is τ -ALUR ⇔ every x
∗
∈ NAX is a τ -strongly exposing functional. Thus, Theorem 2.11 gives a new characterization of τ -ALUR spaces.
As noted earlier, if a finite-codimensional subspace Y of X is proximinal in X, then Y
⊥
⊆ NAX . We now show
that the converse holds in τ -ALUR spaces. It is not difficult to show that the condition Y
⊥
⊆ NAX forces XY to
be reflexive [14, Lemma 2.2]. Thus, our result improves [9, Corollary 3.4].
Theorem 3.2. Let X be a τ -ALUR Banach space and Y be a subspace such that XY is reflexive. Then the following are equivalent
: a Y is proximinal in X;
b Y is τ -strongly proximinal in X; c Y is τ -strongly Chebyshev in X;
d Y
⊥
⊆ NAX.
Proof. By Theorem 2.11, a ⇔ b ⇔ c.
a ⇒ d. Since XY is reflexive, every x
∗
∈ Y
⊥
≃ XY
∗
is norm attaining on XY . Since Y is proximinal in X, x
∗
∈ NAX . Thus Y
⊥
⊆ NAX .
d ⇒ a. Let Y
⊥
⊆ NAX . Since X is τ -ALUR, every x
∗
∈ Y
⊥
≃ XY
∗
is in particular an exposing functional. Since XY is reflexive and XY = Y
⊥ ∗
, it is strictly convex. Let Q : X → XY be the quotient map. If t ∈ S
XY
, there is x
∗
∈ Y
⊥
with x
∗
= x
∗
t = 1 and since XY is strictly convex, {u ∈ S
XY
: x
∗
u = u = 1} = {t}. Now
since x
∗
∈ Y
⊥
⊆ NAX , there is x ∈ X with x = x
∗
x = 1, and thus Qx = t . Hence QB
X
= B
XY
and Y is proximinal in X.
✷
Remark 3.3.
The same proof shows that if Y is a subspace of X such that any x
∗
∈ Y
⊥
is a τ -strongly exposing functional, then Y is τ -strongly Chebyshev.
Let Y be a finite-codimensional proximinal subspace of X. Suppose Y
⊥
has a basis consisting of τ -strongly ex- posing functionals. Does it imply that Y is τ -strongly Chebyshev in X?
We now compare ALUR with some related convexity notions.
Theorem 3.4. For a Banach space X, X is LUR ⇒ X is ALUR ⇒ X is strictly convex and has the KK property. And neither converse implication holds.
Proof. The fact that LUR ⇒ ALUR, but not conversely, has been noted in [2]. Indeed, LUR ⇒ ALUR is clear from
the definitions. And as noted in [2], if X is an infinite-dimensional Banach space with separable dual, then there exists an equivalent norm on X such that X is ALUR but fails to be LUR.
Assume now that X is ALUR. Then X is clearly strictly convex. Let {x
n
}, x ⊆ S
X
be such that x
n
→ x weakly.
Let x
∗
∈ S
X
∗
strongly expose x . Then x
∗
x
n
→ x
∗
x =
1 so that x
n
→ x in norm. Hence X has the KK property.
316 P. Bandyopadhyay et al. J. Math. Anal. Appl. 341 2008 309–317
Let X = c and let T : ℓ
2
→ X be a weakly compact operator with dense range. Define
x
∗
= x
∗ 1
+ T
∗
x
∗ 2
. By [11, Proposition III.2.11], this defines an equivalent dual norm on X
∗
such that c is an M-ideal in ℓ
∞
. This new norm on X
∗
is strictly convex and has the KK property. Now we show that there are points in NAX
∗
that are not even w
∗
-exposing for B
X
∗∗∗
. Indeed X, with the ||| · |||-norm, is an M-ideal in X
∗∗
. So X
∗∗∗
can be decomposed as X
∗∗∗
= X
∗
⊕
1
X
⊥
. By Bishop–Phelps Theorem, there is x
∗∗∗
∈ S
X
∗∗∗
\ X
∗
∪ X
⊥
such that x
∗∗∗
∈ NAX
∗∗
. Then there is x
∗∗
∈ X
∗∗
such that x
∗∗∗
x
∗∗
= |||x
∗∗
||| = 1. Now, we can write x
∗∗∗
= x
∗
+ z
∗∗∗
with x
∗
= 0 ∈ X
∗
, z
∗∗∗
= 0 ∈ X
⊥
and 1 = |||x
∗∗∗
||| = |||x
∗
|||+|||z
∗∗∗
||| . It follows that x
∗∗
x
∗
= |||x
∗
||| , so that x
∗∗
∈ NAX
∗
, but x
∗∗
does not w
∗
-expose x
∗
|||x
∗
||| in B
X
∗∗∗
. ✷
Remark 3.5. Note that the last part of the proof shows that if X is not reflexive and an M-ideal in X
∗∗
, then X
∗
always fails to be wALUR. We conclude this section with the following result.
Theorem 3.6. An ALUR Banach space X is reflexive ⇔ the intersection of any two proximinal hyperplanes is prox- iminal.
Proof.
If X is reflexive, then all subspaces are proximinal. Conversely, let X be ALUR and suppose the intersection of any two proximinal hyperplanes is proximinal. Let
x
∗
, y
∗
∈ NAX . Then ker x
∗
and ker y
∗
are proximinal hyperplanes in X. If Y = ker x
∗
∩ ker y
∗
is proximinal in X, then Y
⊥
⊆ NAX , and therefore, αx
∗
+ βy
∗
∈ NAX for all α, β ∈
R. This implies NAX is a linear subspace of X
∗
. Since X is ALUR and since the set of strongly exposing functionals form a G
δ
set, NAX is a dense G
δ
in X
∗
. Thus, by the Baire Category Theorem, for every x
∗
∈ X
∗
, we have x
∗
+ NAX ∩ NAX = ∅ . Hence NAX −
NAX = X
∗
. Since NAX is a linear subspace of X
∗
, NAX = X
∗
, which, by James’ theorem, implies that X is reflexive.
✷