455
Internat. J. Math. & Math. Sci.
VOL. 15 NO. 3 (1992) 455-464
ON THE LOWER SEMI-CONTINUITY OF THE SET
VALUED METRIC PROJECTION
FOWZI AHMAD SEJEENI
Department of Mathematical Sciences
Umm AI-Qura University
Makkah, Saudi Arabia
(Received November 28, 1988 and in revised form July 30, 1991)
ABSTRACT. The lower semi-continuity of best approximation operators from
Banach lattices on to closed ideals is investigated. Also the existence of
best approximation to sub-function modules of function modules is proved. The
order intersection properties of cells are studied and used to prove the above
results.
KEY WORDS AND PHRASES. Lower semi-continuity, metric projections, continuous
selections,
order
finite
intersection
properties
for
Banach
cells,
lattices,
function modules.
1980 lathematics Subject Clssificaion Code. 41A30, 41A28.
1. INTRODUCTION AND DEFINITIONS.
During
the last 20
years
a
series of
papers
continuity of the set valued metric projection
been concerned
have
with
from normed linear space on to
proximinal linear subspace. Throughout this paper we deal with approximation
of elements of the Banach lattice E by elements of a closed ideal G. For x
we shall denote by
for which [Ix-
d(x,G)
d(x,G)
go[
denote by
PG(X)---
--
inflix- gl] the distance from x to G. Every
f
is called a best approximation
g
G
the set of all best approximation of x
mapping
P
E
2G0
Ilx-gll--by
of’x
in G. We shall
d(x,G)}
elements
of
go
E
e G
{I.I)
G.
The
which associates with each element x of
set
valued
E its {possibly
E on
empty} set of nearest elements in G, is called the metric projection of
to G
{or the metric projection of
E associated with G}.
In recent years, there has been considerable interest in continuous
G with the property that s{x}
E
for every x
E.
PG{x}
mapping s
Such a mapping, if exist, is called a continuous selection for the metric
projection
projection
PG
PG
The available results on continuous selections for the metric
deal primarily with there existence, which follows directly from
according to a result of E. Michael [8].
For set valued mappings, various concepts of continuity are defined as
follows.
the lower semi-continuity of
PG
F. A. SEJEENI
456
DEFINITION 1.1.
E
x
{II)
o U # 0
x
PG
The metric proiection
/
(III}
G
Finally,
x
PG{X}
E
x
st
s open for each open set U in G.
s upper
if
[u.s.c.)
semi-continous
he
is closed for each closed set C in G.
o C
is continos lin the Hausdorff
topology}
metric
if
x imphes
max
d(g,PG{X}):
sup
PG
If
[x
topology iml,lies
g
PG(X}
d(h,eG(X}):
sup
Hausdorff metric topoloCy can
in
Continitv
l.s.c.).
,
is
compact
{.s.c. }.
for
each
x
if G is
Finally,
every closed sphere in a compact set} then
in
PG(Xn)
h
0.
be easily shown to imply
E then the Hausdorff metric
boundedly
compa-t
PG is always
{G
interse(’l s
{u.s.c.} and Hausdorff
metric topology is equivalent to {l.s.c.}.
The
of
metric
projection
For
subspaces.
{l.s.c.}
is
I.
example,
or
{.s.c.}
[12]
Singer
has
only
for
proved
restricted
that
the
class
metric
projection associated with an approximatively compact subset G of a normed
linear
space
E is
{u.s.c.).
Hence,
in
G
may fail
PG
is
{u.s.c.}
if
G is a
But even if G is a linear subspace of
linear subspace of finite dimension.
finite dimension
particular
be {l.s.c.) as A. J. Lazar, P. P. Morris and
D. E. Wulbert have shown in
A subspace G is proximinal if
PG{x)
DEFINITION 1.2. A normed linear lattice is a normed linear space which is
vecr
also a
y
[x
lattice, in which the order and the norm arerelad as follows
implies [x a y[. If the space is complete, it is cled Banh
ttice.
DEFINITION 1.3. A Banh lattice
whenever elements x, y and two collections
(.(xa,ra)t,eh, (.(y,,s,)t
B of
cells are given in E satisfying
(1)
xa
{}
B(x.,r.) (b,Sb)
B(.,.) d
X
y
for each s,b
Yb
for eh
b
B(,)
then
A Banach lattice E has the f.o.i.p. (finite order intersection property)
if the above property holds when the index sets A and B are finite. Also
It
is known that f.o.i.p., the splitting property and f.o.i.p, in the case
BI
JAJ
1 are equivalent
we
now list some examples of Banach lattices with the f.o.i.p.
(I) If E is an (AL)-space, then it has the f.o.i.p.
(2) Every injective Banach lattice {and any closed ideal of it) has the
f.o.i.p.
LOWER SEMI-CONTINUITY
(3) The space
C(x)
457
has the f.o.i.p, if and only if X is Stonian.
{4) Any {AM)-space has the f.o.i.p.
For the proofs and general treatment of injective Banach lattices and Banach
lattices that have the f.o.i.p., we refer the reader to D. Cartwright
[3]
The following fundamental properties of meet, join and the absolute value
will be used freely in the sequel
x v y + x
{1) x + y
(2)
Ix + Yl
(3)
[Ix ly[[ [x-
141
(5)
i6}
7
Ix[
y
^
+
<
lx-x^zl =[xz-zl.
Ixl ^ IY[ 0 if and only if
{x ^ y} + {x ^ z}
x ^ {y + z}
Ix-
y[ =xv
for all x,y,z > 0
y-x^ y.
-
We will prove the following results:
Let E be a Banach lattice with the finite order intersection property,
a closed ideal of E. For each x in E define
Then the set valued real)ping
Let
in T,
F_,
E
<
PG(X}
g
x}
O
PG is
x
,,
,
and
PGlX1
(1.2)
lower semi-continuous.
be a function module and
a
s,b-C(T)-module
of
I,v.
is a Banach lattice with the f.o.i.p, and the fiber G
It has been shown in
IqOTIVATION.
[11]
If for each
is an ideal in
that closed ideals in injective
Banach lattices are always proximinal and the metric projections associated
l.s0Col. These results and the fact that injective
always
with ideals are
Banch lattices have the splitting property lead us to think about the above
results do hold not only in injective Banach lattices but also in Banach
lattices that have the f.o.i.p. The existence of best approximation to ideals
2.
NE’rRIC PROJEe’rIONSo
In order to prove the results stated in the introduction, we need the
following partial results which perhaps are interesting in themselves.
PROPOSI’rION 2.1. Let E be a Bnach lattice, (3 a proxiinal ideal in Eo
Then for each positive element x of E, the following hold;
d(x,G)
Ill
121 x
^
d(x
PGlX) and
^ x PG(x)
g
{3) {f v g)
go
PROOF. {1) Let
PG(x)
PG{x
v g)
).
d(x,G)
(since-
PGlX v
g
V f, g > 0
x v g + x
goll
G).
Ilxvg +
(x^g
(since [ix ^
g)
g
such that f e G and g
^
g
g
PGlX).
PG{x}.
fixed element of
g. Then
go)ll
d(xvg. G)
(2.1)
Similarly,
l[(xvg}- h
llx
together.
for each positive
g)
respectively (h) be an arbitrary but
go}
d(xvg,G)
for each lxsitive g in 13
Write x
Ilx-
g
{x^g
G)
g,
g + h
]
in
G)
goll
(2.2)
d(x,G)
and the result follows from
{2.1) and
{2.2)
458
F.A. SEJEENI
{2)
Ix
{x
-
(x A
[[x-
IIx
g)[[
G). Similarly
[x- (x )J
(since x A g
J(x v g)- gJ
(
x- {x
][x- g[[
IZ)
^
xll
(f v )
Ilx
-< x v
x
!
11
(inc
d(x,G), hence
g
x
g
^
Ix- I
I
which in turn implies that
implies Jl(x v g)-
I-
!
g
t"
^ g
x
and
f
A
PG(X)
Jlx-
(x
g)ll
IIx
gll
d(xVg,G).
g
[’
x-
[f v z) a
then
) impie
xll
d(x,G).
PROPOSITION 2.2. Let E be a Banach lattice, G a closed ideal in E and
an element of E. If
is an element of G and e is a sitive real number such
that II- II
r
where r
d(,G) ), then there is an element I in G
s
%
such that
and IIr +
+
-
* =-
PROF. Define
I- I
.
"I"I- ’"’" II ’
+
(
+
=
" IPl,
(ince
+"-* II)
-" II
B
"
I"I- II
*
s r +
(-
I- I.
.
-
=
.d
II
I,.,_
- q-
llq
Thus, we get
II
’.
*
.Te.,
O)
PROPOSITION 2.3. Let E be a Banh lattice with the f.o.i.p., G a closed
ideal of E and q an element of E. If
is an element of G such that p+ s a
s q- and llq
s r +
then there is an element
in G such that
II- II s r and
( e
,
(for
lal
PelF. Let 0 be a sitive best a approximation of
the exisnce of 0 see F. A. Sejeeni [II]). Now,
such
tha 0
III III
B(II,). B(l"l,r) *
(.i.e
"+)
(3) lPl e B(II,)
and I v O e
Then there exist a o in JIll,IS[ v 0 ]
B(II,) B(ll,r). Define
o A + and
o A
( a
0) and no that,
v
+ -) o A a* + O A
AI:
1 +
[z
hve haL o
G. To see Lhis,
*
t+ + t-.
no
a* lJ, O-
e II
’ l#J
and
PROSITION 4. Let N
closed ideal in
subset of G.
show LhL t
LhL,
JffJ
-
t t-
is Lhe desired elemenL
I’ ’"I
o implies that
#*
*
and
I -,I
#"
a Bah ttice th the f.o.Lp, and G a
Then for eh x in E, the set
{x}
is a nonempty cloud
459
LOWER SEMI-CONTINUITY
PROOF. Let gl respectively
x
gz
10
x
:1:
< x 1.
(since
Ix
PG{Jxl)
Ixl
gl
Igl )
PG{X /) {PG(X ))
in
[g[
and
PGlX).
Plxl
lxl,
REMARK. In the following example
that if g
PG{X), then gv PG{XV)
d{x,)
f-
E
Put g
To see it is closed we may
{g}
g
and
gn
gn
x
A
be
t.
0
belongs to
x-
E
PG{X)
=Cl[oJ{]l,
E be defined by x(t,
Let x
Then the element
(sin(t))*
we will show that it is not always true
Consider the Banach lattice
gl[/2, ]
{sin{t))-.
(since
gl
implies that g e
EXAMPLE.
tg
Jx I.
+ g2 < h
is infinite, Let
g
Ix) converging to g, then
be such that 0 <
PG{lxl) ). Now, let f be an arbit-
hence g
assume with out loss of generality that
sequence in
g
be such that
g- { g pG{ x:l:)
g
h and g
^
Let h in
{g2)
g
G,
defined by
g-
and yet
and the ideal
max
G
{- (t),0
g(t)=- max
g does not belong
G)
0
G
2 is (l.s.c.) if and
PROPOSITION 2.5. The set valued mapping P E
P(x),
only if for each sequence {x
in E converging to x and foreach g
g
there is a sequence
in G with gnf P(xn) and gn converges
PROOF. Assume that P is (l.s.c.), {x n} a sequence in E converging to x
{gn
g
a,d
P{x). The set
of P the set
(y
Uk
B{g, 2-k)
U
P{y)
E
there exists an integer N
[[gn-
such that
g]]
2"k (n Nk).
Now, assume that P is not {l.s.c.)
{ y, E
set
}
Uk$
P(y) U
).
f
and then,
{gn
such
P(xn}
o U #
k
that
gn P(xn)
c
for
and
then for some open set U in G the
is not open. Let x
$
Then by (l.s.c.)
is a neighborhd of x. Hence,
xn k’
Now, We can select a sequence
N k.
each n
{k
G is open in G
,
neighborhd V of x intersects
in a nonempty set
e
}. Let, g
P{x)
pick x
U, and for each n
be such that, each
Rc
is the complement of
in
B{x,2-n)
c.
The
n} converges x, but it is imssible for any sequence {gn} with
gn P{xn) converge g {since U is a neighborhd of g and
THEOREM 2.6. Let E be a Banach lattice with the f.o.i.p, and G a closed
0
G defined by P {x)
ideal in E. Then the set valued mapping
E
2
sequence {x
{g
e
PG(X): g x, ge
PG(X’)}
,
PG
is always (l.s.c.).
PROF. First, we will show that the result holds for positive elements.
For,
g
let
P{x).
Xn be n sitive
let
a
For each
sequence in E
PG{Xn)
hn
then, we have
{I} g
^
x
< g
^
x
(g v h
^
x
x
converging
If
is a
x
{x
0), and
sitive real number
460
F. A. SEJEENI
since
IIx
g
Ix
x
^
<
V
n >
N
[g
Xn,
a x
gn PGlXn}
such that
rn d(xn’G)
and r
,r-r,n
’xx’,n
3max
I +
there ia a g in
] B(g a x ,)
N
(g v h
<
e/ + r + el
where
Then for each n
xll + IIx- gll + IIg
’g-g
gn-
and
x
(g
figs- g
g
).
x
+ el) +
(r
B(xn,rn).
’,
^
Now, for n
ke
Thus we can select a sequence
e
Xn)
figs- Ig
A
{g,}
The sequence
is
the
x II + IIg
be an arbitrary sequence in E converging
x and g an
arbitrary element of
Then by the above there are sitive sequences
and
in G such that
g h
f
g and k
We may assume without loss of generality
desired sequence
Now, let
.
g
{x }
lxl.
tfn}, (hnt
/kn/
that f + h
xna
since
(otherwise
k
kn h. gnV PG (xln
because
(Ix.l.
P
k
gn= gn
that gn
G{In }"
mxn
yl
I=.-
yml
I I g.I
But,
IxnIA kn= k n, then I..- ,I
implies that
1). Now set gn
g: (it is obvious
and
’Io complete the prf, we will show
xn-
kn
k v (f + h
set k
a
inequality holds since
fn FG(Xn) hn
g
x
that
k
f
gn g).
To see this let y be an
while the second one holds
g.
+ g
,.,I l,x.,-
dtxn’G)
i.e.,
gn PG{xn}.
FUNCTION MODULES.
DEFINITION 3.1. Let A be a Banach algebra with a norm
be a Banach space. We say that E is a Banach A-module if
{i) E is a left module over A in the usual algebraic sense;
3.
J A’
aEA, xEE.
Let T be a nonvoid compact Hausdorff space
spaces. The product
.
(Et)tET
and let E
a family of Banach
[[ E t can be thought of as a space of functions on T where
tat
the values of the functions at different points lie
(possibly}
in different
spaces. We will restrict our attention to the subspace
(where
H
t
E
tT
[I
tCT
Et:
=
t).
DEFINITION 3.2. A function module is a triple
of Banach spaces
(the
component
(called
spaces)
such that the following are satisfied:
<
tT
is the norm on the Banach space E
is a nonvoid compact Hausdorff space
x(t)
sup
(R)
IT, (Et)tT E,
the base
and
space),
where T
(Et)tT a family
a closed subspace of
]] E t
taT
46
LOWER SEMI-CONTINU ITY
b’,
I1)
,,ous sc:l:t,"
C{T)-mdule (where C(T) is
valued functions on T) (f.)lt}
is a
the Banach algebra of all contine
fltl.lt} f e C(T),
E.
semi-cont intlous.
13) E
A sub-fuction module is a subspace which is a C{T)-module. A function module
of Baach lattices is a fnction module such that the components E are Banach
and v which are
is closed under the the lattice operations
lattices a,d
(1
defined pointwise
v )(t)
[,
q
and
an element of
(i) the element
.
(t)).
of
a sub-function
E
be a function module in
DEFINITION 3.3. Let
modttle of
{t) v
tT
Then,
is global best approximation of
q
if
tT
{ill the element
is a lal best approximation of a if
LEMMA 3.4. Let
be a function module of Banach lattices and
sub-function module of
each
T
r +
a-
I(t)l
Let
((t))*
and
(qlt))-
in
(lt))-for
each
d(-,G) ).
where r
((t))*
("(t))*
e,
a
is an ideal in E t. Then for
and for each positive real numbers e there is an element
in
such that
t
such that the fiber G
and
2(t)= (=(t))-
2’ then,
IO(tll.
e
Then
i=
tT
.
for each t e T we have
((=(t)l*
a(t))
+
(((t))-
2(t))
I=(tl- I(t)l
l=tl v IO(t)l- ((=(t)l* IO(t)!
l=(t)! v IO(t!- l=(t)l IO(tl
I(t)-
+ (=(t)l-
IO(t)l)
[l=(t)l- lO(t)ll
Hence
II(t)
(t)
.
THEOREM 3.5. Let
function module of
fiber G
II(t)
O(t)
be a function module of Banach lattices and
If for each t
T the space E
a sub-
has the f.o.i.p, and the
is an ideal of E t, then
is proximinal.
e a positive real number and r
PROF. Let
be an element of
d(,G).
((t))
Then by lemma 3.4 there is a
such that IIq- 11
in
r + e and
((t)) for each t e T. Now, by proposition 2.3 there is a gt in G
x
such that
462
Put
F. A. SEJEENI
1/2(l(t)
ft
t
:
then
ftllt <r+
=(t)-
Let
gt),
+
e/2.
a neighborhood of t such that for
U
and
<
we have
U
T
in T be such that
i1
/Uti}l.
lift-l{t}ll
and
t {t) ft
be such that
each s
c/2
Set
l
Ut
and
{fi }t1
’f‘ It‘
the partition of unity subordinate to
then for t z T
l-,
i=
i1
i1
i-I
Similarly
Taking e
<
r+ e/2.
,-,o
e/2.
.
2
(n e 11 we can construct a sequence
I
r + 2
and
n
The second inequality of {3.5} implies that
limit I in
{In}
satisfying
In/1
(In
is Cauchy, and then it has a
The first inequality of {3.5) implies that
e
P;
G{a)
PROPOSITION 3.6. Let T be a compact Hausdorff space, E a Banach lattice
with the f.o.i.p, and G a closed ideal of E. Then C(T,G) contains local and
global best approximation for each f in
C(T,E).
PROOF. The existence of global best approximation follows from theorem 3.5
and the fact that the space
C(T,G)
For local best approximation
we
is the set of all
(where
{1.2) ). Then set valued mapping
0
{l.s.c.). Let s
continuous and
f is desired element of G
s
PG(f{t))
PGis
is an ideal in the Banach lattice
define
P
T
2
G
by P{t)
C(T,E).
P(f(t))
best approximation of f(t) in G satisfying
P
is {l.s.c.)
since
P
0
f),
P,
then
(PG
be a continuous selection of
f
is
g
Finally, we conclude this paper with the observation that in some cases
global best approximation always exists and yet the set valued mapping P
defined above admits no continuous selection at all as we will see in the
following.
LOWER SEMI-CONTINUITY
EXAMPLE. Let
[0,I]
[0,1/2].
functions on
vanish on
{t
G
T)
/
E=
with
C{[0,1],)
[[f[]
If we take
{the space of all continuous real valued
suplf{t}l},
tET
f E
463
E
the space of all f in
to be the
constant
E
function
which
f{t}
1
Then we have the following
{0}
e
fr t
for t
[0’1/2]
(1/2,1]
and
P{t,
Thus P is a single valued discontinuous function
{0} if
{1} if
on [0,1]
t e
t
[0,1/2]
(1/2,1]
which admits no
continuous selection at all.
REFERENCES
I. Behrends, E. M-structure and the Banach-Stone Theorem, Lecture Notes in
Mathematics, # 736, Springer Verlag, NewYork 1979.
2. Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloquium Publ.Vol. 25,
Providence, R. I., 1940.
3. Cartwright, D., Extensions of positive operators between Banach lattices,
Memoirs, Amer. Math. Soc. 164 {1975} PP 1-48.
4. Fakhoury,H.,Existence d’une projection continue de Meilleure approximation
dans certains espaces de Banach,J.Math. Pures et Appl. 53 {1974} PP 1-16.
5.,
Gierz, G., Injective Banach lattices with strong order unit, Pacific J. of
Math. Vol. 110 No. 2 {1984} PP 297-305
6. Haydon, R., Injective Banach lattices, Math. Z. 156 (1977).
7.
Lazar, A.J., Morris, P.D., Wulbert, D.E., Continuous selection for metric
projections, J. Func. Anal. 3 {1967} PP 193-216.
8. btichael,E., Continuous selection Ann. Math. 63 {1954) PP361-382
9. Rieffel, M.A., Induced Banach representation of Banach algebra and locally
compact groups, J. Func. Anal. 1 {1967} PP 443-491
10. Schaefer, H.H., Banach lattices and positive operators, Grundlehren der
Math. Band 215, Springer Verlag, NewYork 1974.
U.C.R.
11. Sejeeni, F.A., Best approximation in C{T}-modules, Dissertation
Riverside Dce. 1986.
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