ON THE LOWER SEMI-CONTINUITY OF THE SET VALUED METRIC PROJECTION

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


INTRODUCTION AND DEFINITIONS.
During the last 20 years a series of papers have been concerned with continuity of the set valued metric projection 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 E we shall denote by d(x,G) inflix-gl] the distance from x to G. Every go e G for which [Ix-go[ d(x,G) is called a best approximation of'x in G.We shall denote by PG(X)---f g G Ilx-gll---d(x,G)} {I.I) the set of all best approximation of x by elements of G.The set valued mapping P E --2G0 which associates with each element x of E its {possibly empty} set of nearest elements in G, is called the metric projection of E on to G {or the metric projection of E associated with G}.
In recent years, there has been considerable interest in continuous mapping s E G with the property that s{x} PG{x} for every x E.
Such a mapping, if exist, is called a continuous selection for the metric projection PG The available results on continuous selections for the metric projection PG deal primarily with there existence, which follows directly from the lower semi-continuity of PG according to a result of E. Michael [8].
For set valued mappings, various concepts of continuity are defined as follows.
DEFINITION 1.1.Continitv in Hausdorff metric topoloCy can be easily shown to imply l.s.c.).If PG [x is compact for each x in E then the Hausdorff metric topology iml,lies {.s.c.}.Finally, if G is boundedly compa-t {G interse('l s every closed sphere in a compact set} then PG is always {u.s.c.} and Hausdorff metric topology is equivalent to {l.s.c.}. The metric projection is {l.s.c.} or {.s.c.}only for restricted class of subspaces.For example, I. Singer [12] has proved that the metric projection associated with an approximatively compact subset G of a normed linear space E is {u.s.c.).Hence, in particular PG is {u.s.c.} if G is a linear subspace of finite dimension.
But even if G is a linear subspace of finite dimension G may fail be {l.s.c.) as A. J.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 JAJ BI 1 are equivalent we now list some examples of Banach lattices with the f.o.i.p.
(3) The space C(x) has the f.o.i.p, if and only if X is Stonian.
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 {1) x + y x v y + x ^y (2) Ixl ^IY[ 0 if and only if i6} x ^{y + z} {x ^y} + {x ^z} for all x,y,z > 0 7 Ixy[ =xv 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 x} g PG(X} < x , and -PGlX 1 (1.2) O Then the set valued real)ping PG is lower semi-continuous.
Let F_, be a function module and ,, a s,b-C(T)-module of I,v.If for each in T, E is a Banach lattice with the f.o.i.p, and the fiber G is an ideal in IqOTIVATION.It has been shown in [11] that closed ideals in injective Banach lattices are always proximinal and the metric projections associated with ideals are always l.s0Col.These results and the fact that injective 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; for each lxsitive g in 13 121 x ^g PGlX) and g PGlX v g) for each positive g PGlX).
{3) {f v g) ^x PG(x) V f, g > 0 such that f e G and g PG{x}.PROOF.{1) Let go respectively (h) be an arbitrary but fixed element of PG(x) PG{x v g) ).Write x x v g + x ^g g.Then d(x,G) Ilx-goll Ilxvg + (x^g g go)ll d(xvg.G) ) and the result follows from {2.1) and {2.2) together.
(t" g !I-g ! (inc g [' ) impie 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 such that + s % and IIr + .
Thus, we get llq q-B s r + .
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 PelF.Let 0 be a sitive best a approximation of lal such tha 0 (for the exisnce of 0 see F. A. Sejeeni [II]).Now, [z B(II,).B(l"l,r) * (.i.e III III " + ) (3) lPl e B(II,) * a* lJ, O-' l#J and JffJ o implies that #* * and #" PROSITION 4. Let N a Bah ttice th the f.o.Lp, and G a closed ideal in Then for eh x in E, the set {x} is a nonempty cloud subset of G.
PROOF.Let gl respectively {g2) in PG{X /) {PG(X )) be such that 0 < gl x 10 gz < x 1.Let h in PG{Jxl) be such that g + g2 < h Jx I. Put g :1: g pG{ x:l:) x ^h and g g g-{ and [g[ PG {lxl) ).Now, let f be an arbit- (since Ix gl Ixl Igl ) hence g PGlX).To see it is closed we may assume with out loss of generality that Plxl is infinite, Let {g} be g x sequence in Ix) converging to g, then g and gn A gn d{x,) implies that g e lxl, REMARK.In the following example we will show that it is not always true that if g PG{X), then gv PG{X V) EXAMPLE.Consider the Banach lattice E =Cl[oJ{]l, and the ideal G tg f-E gl[/2, ] 0 t.Let x E be defined by x(t, max {-(t),0 {sin{t))-.Then the element g G, defined by g(t)=-max (sin(t))* belongs to PG{X) and yet g- g does not belong (since x-0 G) PROPOSITION 2.5.The set valued mapping P E 2 G is (l.s.c.) if and only if for each sequence {x in E converging to x and foreach g P(x), there is a sequence {gn in G with gnf P(xn) and gn converges g PROOF.Assume that P is (l.s.c.), {x n} a sequence in E converging to x a,d g P{x).The set U B{g,2-k) G is open in G {k ).Then by (l.s.c.) of P the set Uk (y E P{y) Uk$ } is a neighborhd of x.Hence, there exists an integer N such that xn f k' and then, P(xn} o U k # for each n N k.Now, We can select a sequence {gn such that gn P(xn) and [[gn-g]] 2"k (n Nk).Now, assume that P is not {l.s.c.) then for some open set U in G the set { y, E P(y) U $ is not open.Let x be such that, each neighborhd V of x intersects c in a nonempty set Rc is the complement of }.Let, g e P{x) U, and for each n , pick x in B{x,2-n) c.The sequence {x 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 2 G ideal in E. Then the set valued mapping PG E defined by P {x) {g e PG(X): g x, ge PG(X')} is always (l.s.c.).
PROF.First, we will show that the result holds for positive elements.
For, let Xn be a sitive sequence in E converging x {x 0), and g P{x).For each n , let hn PG{Xn) If is a sitive real number then, we have {I} g ^x < g ^x (g v h ^x x since IIx g ^x Ix xll + IIx-gll + IIg g ^x < e/ + r + el < I + (r + el) + V n > N where rn d(xn'G) and r Then for each n N there ia a g in [g Xn, (g v h a x ] B(g a x ,) B(xn,rn).Now, for n ke 3max 'x-x',n ,r-r,n 'g-g x ', ).Thus we can select a sequence such that gn PGlXn} and gn-(g A Xn) e The sequence {g,} is the desired sequence since figs-g g figs-Ig x II + IIg Now, let {x } be an arbitrary sequence in E converging x and g an arbitrary element of lxl.Then by the above there are sitive sequences tfn}, (hnt and /kn/ in G such that fn FG(Xn) hn kn f g h g and k g We may assume without loss of generality that f + h k (otherwise set k k v (f + h 1).Now set gn x k f . xna kn a h.gn V PG (xln and gn= gn g: (it is obvious that gn g).
'Io complete the prf, we will show that gn G{In }" To see this let y be an inequality holds since x n-yl mxn yml while the second one holds because k P (Ix.l.But, I I g.I g. + g IxnIA kn= k n, then I..-,I ,.,I l,x.,- implies that I=.dtxn'G) i.e., gn PG{xn}.
3. FUNCTION MODULES.DEFINITION 3.1.Let A be a Banach algebra with a norm J A' and let E 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; aEA, xEE.
Let T be a nonvoid compact Hausdorff space (Et)tE T a family of Banach spaces.The product [[ 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 .t is the norm on the Banach space E t).DEFINITION 3.2.A function module is a triple IT, (Et)tT E, where T is a nonvoid compact Hausdorff space (called the base space), (Et)t T a family of Banach spaces (the component spaces) and a closed subspace of ]] E t taT such that the following are satisfied: I1) b', is a C{T)-mdule (where C(T) is the Banach algebra of all contin- ,,ous sc:l:t," valued functions on T) (f.)lt} fltl.lt}f e C(T), e E.
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 lattices a,d is closed under the the lattice operations and v which are defined pointwise (1 v )(t) {t) v (t)).tT Let 2' then, 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)! + (=(t)l-IO(t)l) l=(t)!v IO(t!-l=(t)l IO(tl [l=(t)l-lO(t)ll I(t)- II-II- O(t) THEOREM 3.5.Let be a function module of Banach lattices and a subfunction module of .If for each t T the space E has the f.o.i.p, and the fiber G is an ideal of E t, then is proximinal.
PROF.Let be an element of e a positive real number and r d(,G).Then by lemma 3.4 there is a in such that IIq-11 r + e and ((t)) x ((t)) for each t e T. Now, by proposition 2.3 there is a gt in G such that EXAMPLE.Let E= C{[0,1],) {the space of all continuous real valued functions on [0,I] with [[f[] suplf{t}l}, the space of all f in E which tET vanish on [0,1/2].If we take f E E to be the constant function f{t} 1 {t T) Then we have the following G / {0} fr t [0'1 /2] and P{t, {0} if t e [0,1/2] e for t (1/2,1] {1} if t (1/2,1] Thus P is a single valued discontinuous function on [0,1] which admits no continuous selection at all.
, and q an element of .