Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 890598, 10 pages
doi:10.1155/2011/890598
Research Article
On Simultaneous Farthest Points in L∞ I, X
Sh. Al-Sharif1 and M. Rawashdeh2
1
2
Mathematics Department, Yarmouk University, Irbid 21163, Jordan
Department of Mathematics and Statistics, Jordan University of Science and Technology,
Irbid 22110, Jordan
Correspondence should be addressed to Sh. Al-Sharif, sharifa@yu.edu.jo
Received 20 December 2010; Accepted 23 May 2011
Academic Editor: Siegfried Gottwald
Copyright q 2011 Sh. Al-Sharif and M. Rawashdeh. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Let X be a Banach space and let G be a closed bounded subset of X. For x1 , x2 , . . . , xm ∈ X m , we
set ρx1 , x2 , . . . , xm , G sup{max1≤i≤m xi − y : y ∈ G}. The set G is called simultaneously remotal
if, for any x1 , x2 , . . . , xm ∈ X m , there exists g ∈ G such that ρx1 , x2 , . . . , xm , G max1≤i≤m xi − g.
In this paper, we show that if G is separable simultaneously remotal in X, then the set of ∞Bochner integrable functions, L∞ I, G, is simultaneously remotal in L∞ I, X. Some other results
are presented.
1. Introduction
Let X be a Banach space and G a bounded subset of X. For x ∈ X, set ρx, G sup{x −
y : y ∈ G}. A point g0 ∈ G is called a farthest point of G if there exists x ∈ X such that
x − g0 ρx, G. For x ∈ X, the farthest point map FG x {g ∈ G : x − g ρx, G}, that
is, the set of points of G farthest from x. Note that this set may be empty. Let RG, X {x ∈
φ}. We call a closed bounded set G remotal if RG, X X and densely remotal
X : FG x /
if RG, X is a norm dense in X. The concept of remotal sets in Banach spaces goes back to
the sixties. However, almost all the results on remotal sets are concerned with the topological
properties of such sets, see 1–4. Remotal sets in vector valued continuous functions was
considered in 5. Related results on Bochner integrable function spaces, Lp I, X, 1 ≤ p ≤ ∞,
are given in 6–8.
The problem of approximating a set of points x1 , x2 , . . . , xm simultaneously by a point
g farthest point in a subset G of X can be done in several ways, see 9. Here, we will use
the following definition.
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International Journal of Mathematics and Mathematical Sciences
Definition 1.1. Let G be a closed bounded subset of X. A point g ∈ G is called a simultaneous
farthest point of x1 , x2 , . . . , xm ∈ X m if
ρx1 , x2 , . . . , xm , G sup max xi − h max xi − g .
h∈G 1≤i≤m
1.1
1≤i≤m
We call a closed bounded set G of a Banach space X simultaneously remotal if each m-tuple
x1 , x2 , . . . , xm ∈ X m admits a farthest point in G and simultaneously densely remotal if the
φ}, where
set of points RG, X m {x1 , x2 , . . . , xm ∈ X m : FG x1 , x2 , . . . , xm /
FG x1 , x2 , . . . , xm g ∈ G : ρx1 , x2 , . . . , xm , G max xi − g 1≤i≤m
1.2
is norm dense in X m .
Clearly, if m 1, then simultaneously remotal is precisely remotal.
In this paper we consider the problem of simultaneous farthest point for bounded sets
of the form L∞ I, G in the Banach space L∞ I, X, where X is a Banach space.
Throughout this paper, X is a Banach space, G is a closed bounded subset of X and
L∞ I, X, the space of all X-valued essentially bounded functions on the unit interval I. For
f ∈ L∞ I, X, we set f∞ ess sup{fs : s ∈ I}. For G ⊂ X, we set L∞ I, G {f ∈
L∞ I, X : fs ∈ G, almost all s ∈ I}.
2. Distance Formula
The farthest distance formula is important in the study of farthest point. In this section, we
compute the ∞-farthest distance from an element f ∈ L∞ I, X to a bounded set L∞ I, G. We
begin with the following proposition.
Proposition 2.1. Let f1 , f2 , . . . , fm , then max1≤i≤m fi ∞ = ess sup max1≤i≤m fi t.
Proof. For 1 ≤ i ≤ m,
fi ess supfi t ≤ ess sup max fi t ≤ max fi .
∞
∞
1≤i≤m
1≤i≤m
2.1
Hence, max1≤i≤m fi ∞ ess sup max1≤i≤m fi t
Theorem 2.2. Let X be a Banach space and let G be a closed bounded subset of X. If a function
Φ : I → R defined by Φt ρf1 t, f2 t, . . . , fm t, G, where f1 , f2 , . . . , fm ∈ L∞ I, X, then
Φ ∈ L∞ I and
ρ f1 , f2 , . . . , fm , L∞ I, G Φ∞ sup max fi − g ∞ .
g∈L∞ I,G1≤i≤m
2.2
Proof. Let f1 , f2 , . . . , fm ∈ L∞ I, X. Being strongly measurable, there exist m sequences of
simple functions fin , 1 ≤ i ≤ m such that fin t − fi t → 0 as n → ∞ for almost all
International Journal of Mathematics and Mathematical Sciences
3
mn
j
t ∈ I. We may write fjn i1 χAi,n xin . Since ρx1 , x2 , . . . , xm , G is a continuous function
m
of x1 , x2 , . . . , xm ∈ X , the inequality
ρ f1n t, f2n t, . . . , fmn t, G − ρ f1 t, f2 t, . . . , fm t, G ≤ max fin t − fi t −→ 0
1≤i≤m
2.3
implies that
ρ f1n t, f2n t, . . . , fmn t, G − ρ f1 t, f2 t, . . . , fm t, G −→ 0.
2.4
Set Φn t ρf1n t, f2n t, . . . , fmn t, G. Then,
Φn t sup max fkn t − g g∈G 1≤k≤m
mn
k
sup max χAin t xin − g 1≤k≤m
g∈G
i1
mn
i1
2.5
k
χAin t sup max xin − g .
g∈G 1≤k≤m
So Φn is a simple function for each n and Limn → ∞ Φn t−Φt 0 for almost all t ∈ I. Hence
Φ is measurable. Furthermore, for each w ∈ L∞ I, G,
max fi − w∞ max ess sup fi t − wt
1≤i≤m
1≤i≤m
ess sup max fi t − wt ,
1≤i≤m
Proposition 2.1
≤ ess sup sup max fi t − g g∈G 1≤i≤m
2.6
ρ f1 t, f2 t, f3 t, . . . , fm t, G ∞
Φ∞ .
Thus,
Φ∞ ≥ ρ f1 , f2 , . . . , fm , L∞ I, G .
2.7
To prove the reverse inequality. Let > 0 be given, since countably valued functions are
dense in L∞ I, X, there exist countably valued functions γ1 , γ2 , . . . , γm in L∞ I, X such that
∞
j
fi −γi < , 1 ≤ i ≤ m. We may write γj ∞
i1 χAi xi , as in 10. We may assume
i1 χAi 1,
μAi > 0 for all i. For each i, Choose hi ∈ G such that
k
1
2
m
max xi − hi > ρ xi , xi , . . . , xi , G − .
1≤k≤m
2.8
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International Journal of Mathematics and Mathematical Sciences
Now, set g ∈ L∞ I, G as gs ∞
i1
χAi shi . The inequality
γj − g ≤ γj − fj fj − g ∞
∞
∞
2.9
max γj − g ∞ ≤ max fj − g ∞ .
2.10
implies
1≤j≤m
1≤i≤m
Further,
max fj − g ∞ ≥ max γj − g ∞ − 1≤j≤m
1≤j≤m
max ess sup γj t − gt − 1≤j≤m
∞
j
ess sup max χAi t xi − hi − ,
1≤j≤m
i1
ess sup max
1≤j≤m
ess sup
∞
i1
≥ ess sup
∞
i1
∞
i1
Proposition 2.1
j
χAi txi − hi − 2.11
j
χAi t max xi − hi − 1≤j≤m
1
2
m
χAi t ρ xi , xi , . . . , xi , G − 2
ess sup ρ γ1 t, γ2 t, . . . , γm t, G − 2.
For 1 ≤ j ≤ m and a ∈ G, the inequality
fj t − a ≤ fj t − γj t γj t − a
2.12
implies
ρ f1 t, f2 t, . . . , fm t, G ≤ max fj t − γj t ρ γ1 t, γ2 t, . . . , γm t, G .
1≤j≤m
2.13
Therefore,
⎞
⎛ ρ f1 t, f2 t, . . . , fm t, G
⎠ − 2
max fj − g ∞ ≥ ess sup⎝
−max γj t − fj t
1≤j≤m
1≤j≤m
≥ Φ∞ − ess sup max γj t − fj t − 2
1≤j≤m
≥ Φ∞ − 3.
2.14
International Journal of Mathematics and Mathematical Sciences
5
Hence, ρf1 , f2 , . . . , fm , L∞ I, G 3 ≥ Φ∞ . Since was arbitrary, we have Φ∞ ρf1 , f2 , . . . , fm , L∞ I, G.
Corollary 2.3. Let g be a strongly measurable function from I to a closed bounded subset G of
a Banach space X, and f1 , f2 , . . . , fm ∈ L∞ I, X. If gt is a simultaneous farthest point of
f1 t, f2 t, . . . , fm t in G, then g is a simultaneous farthest point of f1 , f2 , . . . , fm in L∞ I, G.
Proof. By assumption, max1≤i≤m fi t − gt ρf1 t, f2 t, . . . , fm t, G for almost t ∈ I.
Since G is bounded, it follows that g ∈ L∞ I, G and
ess sup max fi t − gt ess sup ρ f1 t, f2 t, . . . , fm t, G .
2.15
1≤i≤m
Theorem 2.2 and Proposition 2.1 implies that
max fi − g ∞ ess sup max fi t − gt ρ f1 , f2 , . . . , fm , L∞ I, G
1≤i≤m
1≤i≤m
2.16
and g is a simultaneous farthest point of f1 , f2 , . . . , fm in L∞ I, G.
3. Remotal Sets in L∞ I, X
In this section, we raise the question: if G is a simultaneously remotal set in X, is L∞ I, G
simultaneously remotal in L∞ I, X? We get a positive answer to this question in the case
that G is a separable simultaneously remotal subset of X or in the case that Span G is finite
dimensional subspace of X. We begin with the following theorem.
Theorem 3.1. If G is simultaneously densely remotal in X, then L∞ I, G is simultaneously densely
remotal in L∞ I, X.
Proof. Let f1 , f2 , . . . , fm ∈ L∞ I, X. Then, there exist γ1 , γ2 , . . . , γm simple functions such that
i
fi − γi ∞ < /2, 1 ≤ i ≤ m. Now, we can write without lost of generality, γi nk1 χAk xk .
1
2
m
Since G is simultaneously densely remotal, then there exist an m-tuple yk , yk , . . . , yk i
i
and gk ∈ G such that max1≤i≤m xk − yk < /2 and
i
i
1
2
m
max yk − gk ρ yk , yk , . . . , yk , G sup max yk − g .
1≤i≤m
Set φ n
k1
g∈G 1≤i≤m
χAk gk and hi n
k1
i
χAk yk . Then,
max hi − φ∞ max ess suphi t − φt
1≤i≤m
1≤i≤m
n
i
ess sup max χAk t yk − gk ,
1≤i≤m
k1
Proposition 2.1
3.1
6
International Journal of Mathematics and Mathematical Sciences
ess sup max
1≤i≤m
≥ ess sup max
1≤i≤m
n
k1
n
k1
i
χAk tyk − gk i
χAk tyk − g 3.2
for every g ∈ G. In particular, for any w ∈ L∞ I, G, using Proposition 2.1,
max hi − φ∞ ≥ ess sup max hi t − wt
1≤i≤m
1≤i≤m
≥ max ess suphi t − wt max hi − w∞ ,
1≤i≤m
3.3
1≤i≤m
Hence, φ is a farthest point from the m-tuple h1 , h2 , . . . , hm . But
n
i
i
y − xk max hi − γi ∞ max ess sup
χ
k1 Ak k
1≤i≤m
1≤i≤m
max ess sup
1≤i≤m
k1
ess sup max
1≤i≤m
ess sup
n
n
k1
n
k1
i
i χAk yk − xk 3.4
i
i χAk yk − xk ,
Proposition 2.1
i
i χAk max yk − xk ≤ .
1≤i≤m
2
Hence,
max fi − hi ∞ ≤ max fi − γi max γi − hi ≤ .
1≤i≤m
1≤i≤m
1≤i≤m
2 2
3.5
This complete the proof of the theorem.
For a remotal set G ⊆ X, the map T : X m → 2G defined by
T x1 , x2 , . . . , xm Fx1 , x2 , . . . , xm , G
g ∈ G : ρx1 , x2 , . . . , xm , G max xi − g
3.6
1≤i≤m
is a multivalued map in general. Hence for any f ∈ L∞ I, X, the map T ◦ f is a multivalued
map from I into G.
Before proceeding we remind the reader for some facts regarding mult-valued maps.
For a Banach space X and a measurable space I a function f : I → X is said to be strongly
measurable if it is the pointwise limit of a sequence of simple functions almost everywhere.
International Journal of Mathematics and Mathematical Sciences
7
On the other hand f is said to measurable in the classical sense if f −1 K is measurable in I
for any closed set K in X, see 10. A multivalued function T : I → X is said to measurable
in the classical sense if T −1 K is measurable in I for any closed set K in X, here T −1 K {t ∈
I : T t ∩ K / φ}. A measurable in the classical sense g may be extracted from a measurable
multivalued function T : I → X where X is a separable Banach space provided that T t is a
closed subset of X for each t ∈ I and such that gt ∈ T t for each t ∈ I, see 11, page 289.
Theorem 3.2. Let G be a closed bounded simultaneously remotal in X such that spanG is finite
dimensional subspace of X, then L∞ I, G is simultaneously remotal in L∞ I, X.
Proof. First, we will prove that T is a closed valued map. If T x1 , x2 , . . . , xm is finite set,
then it is closed. If T x1 , x2 , . . . , xn is not finite, let y ∈ T x1 , x2 , . . . , xm , then there exists
yn ∈ T x1 , x2 , . . . , xm such that yn → y. This implies
max xi − g ≤ max xi − yn ,
1≤i≤m
1≤i≤m
3.7
for every g ∈ G. Taking the limit as n → ∞, we get max xi − g ≤ max xi − y for every
g ∈ G, and this implies that y ∈ T x1 , x2 , . . . , xm and T is a closed multivalued map.
To prove that T is measurable in the classical sense let B be any closed subset of G.
n
n
n
If x1 , x2 , . . . , xm ∈ T −1 B, then there exists x1 , x2 , . . . , xm ∈ T −1 B that converges to
n
n
n
n
n
n
x1 , x2 , . . . , xm . Since x1 , x2 , . . . , xm ∈ T −1 B, then T x1 , x2 , . . . , xm ∩ B /
φ. Choose
n
n
n
yn ∈ T x1 , x2 , . . . , xm ∩ B. Then, yn has a convergent subsequence ynk → y being a
sequence in a bounded closed subset of a finite dimensional space. But
n n max xi k − ynk ≥ max xi k − g 1≤i≤m
1≤i≤m
3.8
for every g ∈ G. Taking the limit as nk → ∞, we get
max xi − y ≥ max xi − g 1≤i≤m
1≤i≤m
3.9
for every g ∈ G. Hence y ∈ T x1 , x2 , . . . , xm ∩ B / ∅. Therefore x1 , x2 , . . . , xm ∈ T −1 B and
−1
T B is closed. Hence, T is measurable and if γ is the vector valued map γ : I → X m ,
γt f1 t, f2 t, . . . , fm t, then T ◦ γ is a measurable closed multivalued map. By Theorem
6.6.4 in 11, page 289, T ◦ γ has a measurable selection say g. Further,
max fi t − gt ≥ max fi t − ht,
1≤i≤m
1≤i≤m
3.10
for every h ∈ L∞ I, G and so
ess sup max fi t − gt max ess supfi t − gt
1≤i≤m
1≤i≤m
≥ max ess supfi t − ht.
1≤i≤m
3.11
8
International Journal of Mathematics and Mathematical Sciences
Hence,
max fi − g ∞ ≥ max fi − h∞ ,
1≤i≤m
3.12
1≤i≤m
for every h ∈ L∞ I, G and L∞ I, G is remotal in L∞ I, X.
Finally, following the steps in the prove of Theorem 3.6 in 6, we proof the following
main result in the paper.
Theorem 3.3. Let G be a separable simultaneously remotal subset of X. Then L∞ I, G is
simultaneously remotal in L∞ I, X.
Proof. Let f1 , f2 , . . . , fm ∈ L∞ I, Xm . Using Corollary 2.3, it is sufficient to show that there
exists a measurable function g defined on I such that gt is a simultaneous farthest point of
f1 t, f2 t, . . . , fm t in G. Since f1 , f2 , . . . , fm are strongly measurable. We may assume that
f1 I, f2 I, . . . , fm I are separable sets in X. So there exist {Ini }∞
n1 a countable partition of I
such that diamfi Ini < 1/2, for each i, 1 ≤ i ≤ m, for all n; where
diamS sup x − y : x, y ∈ S .
3.13
m
Consider the partition Ik1 k2 ···km m
i1 {Ini }, for 1 ≤ n < ∞.
i1 Iki , where Ik1 , Ik2 , . . . , Ikm ⊆
.
Then diamfi Ik1 k2 ···km < 1/2. For simplicity write Ik1 k2 ···km as {In }∞
n1 For each t ∈ I, let g0 t
be a simultaneous farthest point from f1 t, f2 t, . . . , fm t in G. Define the map g0 from I
into G by g0 t is a simultaneous farthest point from f1 t, f2 t, . . . , fm t. Apply Lemma
3.1 in 6 with 1/2 and I In A, we get countable partitions in each In and therefore
countable partition in the whole of I in measurable sets {En }∞
n1 and a sequence of subsets
such
that
{An }∞
n1
An ⊆ En ,
μ∗ An μEn ,
1
1
diam g0 An < and diam fi En < ,
2
2
1 ≤ i ≤ m.
3.14
Repeat the same argument in each En with 1/22 , I En , and A An . For each n, we get a
countable partition {En,k : 1 ≤ k < ∞} of En in measurable sets and a sequence {An,k : 1 ≤
k < ∞} of subsets of I such that
An,k ⊆ En,k ∩ An ,
μ∗ An,k μ En,k ,
1
1
diam g0 An,k < 2 and diam fi En,k < 2 ,
2
2
i 1, 2, . . . , m.
3.15
Now, we will use mathematical induction for each n; let Δn be the set of n-tuples of natural
numbers and Δ {Δn : 1 ≤ n < ∞}. On this Δ consider the partial order defined by
m1 , m2 , . . . , mi < n1 , n2 , . . . , nj if and only if i ≤ j and mk nk , k 1, 2, . . . , i. Then by
induction for each n, we can find a countable partition {Eα : α ∈ Δn } of I of measurable sets
International Journal of Mathematics and Mathematical Sciences
9
and a collection {Aα : α ∈ Δn } of subsets of I such that:
1 Aα ⊆ Eα and μ∗ Aα μEα ,
2 Eβ ⊆ Eα and Aβ ⊆ Aα if α ≤ β,
3 diamfi Eα < 1/2n for i 1, 2, . . . , m and diamg0 Aα < 1/2n for α ∈ Δn .
We may assume Aα /
φ, for all α. For each α ∈ Δ, let tα ∈ Aα and define gn from I into
G by gn t α∈Δn χEα tg0 tα . Then, for each t ∈ I and n ≤ m, we have
gn t − gm t χEα tg0 tα −
χEβ tg0 tβ α∈Δn
β∈Δm
≤
χEβ t g0 tα − g0 tβ β∈Δm
g0 tα − g0 tβ χ
≤
Eβ
3.16
β∈Δm
≤
1
.
2n
Hence, gn t is a Cauchy sequence in G for all t ∈ I. Therefore gn t is a convergent
sequence. Let g : I → G be defined to be the pointwise limit of gn . Since gn is strongly
measurable for each n, we have g is strongly measurable. Further for t ∈ I, 1 ≤ n < ∞, and
t ∈ Eα for some α ∈ Δn , we have
max fi t − gn t max fi t − g0 tα 1≤i≤m
1≤i≤m
≥ max fi tα − g0 tα − fi t − fi tα 1≤i≤m
1
≥ max fi tα − g0 tα − n 1≤i≤m
2
3.17
1
≥ ρ f1 tα , f2 tα , . . . , fm tα , G − n .
2
For 1 ≤ i ≤ m, the inequality
fi t − a ≤ fi t − fi tα fi tα − a ≤ 1 fi tα − a
n
2
3.18
1
ρ f1 t, f2 t, . . . , fm t, G ≤ n ρ f1 tα , f2 tα , . . . , fm tα , G .
2
3.19
implies that
10
International Journal of Mathematics and Mathematical Sciences
Therefore,
1
max fi t − gn t ≥ ρ f1 t, f2 t, . . . , fm t, G − n−1 .
2
1≤i≤m
3.20
Taking limit as n → ∞, we get
ρ f1 t, f2 t, . . . , fm t, G max fi t − gt.
1≤i≤m
3.21
This completes the proof of the theorem.
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