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Best approximative properties of exposed faces of [l.sub.1].

1 Introduction

Throughout X denotes a real Banach space, X*, the dual of X, [B.sub.X] = {x [member of] X : [parallel]x[parallel] [less than or equal to] 1} and [S.sub.X] = {x [member of] X : [parallel]x[parallel] = 1}, the unit sphere of X.

For any x in X and a subset C of X, d(x, C) denotes the distance of x from C and the set [P.sub.C] (x) of best approximations to x from C is given by

[P.sub.C](x) = {y [member of] C : [parallel]x - y[parallel] = d(x, C)}.

If [P.sub.C](x) is a non-empty set for each x in X, we say the subset C is proximinal in X. For any [delta] > 0 we set

[P.sub.C](x,[delta]) = {z [member of] C : [parallel]x - z[parallel] < d(x,C) + [delta]}.

We say a proximinal set C of a normed linear space X is strongly proximinal if for each x in X and [epsilon] > 0, there exists [delta] > 0 such

that sup{d(z, [P.sub.C](x)) : z [member of] [P.sub.C](x,[delta])} < [epsilon].

Let X and Y be normed linear spaces. We say F is a set valued map from X into Y if F is a map from X into the class of all non-empty subsets of Y and in this case, we write F : X [right arrow] Y is a set valued map. Let [x.sub.0] be in X. The set valued map F is said to be

1. lower semi-continuous at [x.sub.0] if, given [epsilon] > 0 and z in F([x.sub.0]), there exists [delta] = [delta]([epsilon], z) > 0 such that the set B(z, [epsilon]) [intersection] F(x) is non-empty, for any x in B([x.sub.0], [delta]). If [delta] can be chosen to be independent of z in F([x.sub.0]) in the above definition, that is, given [epsilon] > 0, there exists [delta] > 0 such that the set B(z, [epsilon]) [intersection] F(x) is non-empty, for any x in B([x.sub.0], [delta]) and any z in F([x.sub.0]), then F is said to be lower Hausdorff semi-continuous at [x.sub.0] .

2. upper semi-continuous at [x.sub.0] if given any open neighbourhood U of zero in X, there exists [delta] > 0 such that

F(x) [epsilon] F([x.sub.0]) + U

for each x in B([x.sub.0], [delta]). Replacing the arbitrary open set U by an open ball in the above, yields the notion of upper Hausdorff semi-continuity. More precisely, the map F is upper Hausdorff semi-continuous at [x.sub.0], if given [epsilon] > 0 there exists [delta] > 0 such that

F(x) [epsilon] F([x.sub.0]) + [??]Bx for each x in B([x.sub.0], [delta]).

The set valued map F is lower (upper, lower Hausdorff, upper Hausdorff) semi-continuous on X if it is lower (upper, lower Hausdorff, upper Hausdorff) semi-continuous at each point of X and is called Hausdorff metric continuousif it is both lower and upper Hausdorff semi-continuous.

We observe that upper semi continuity implies upper Hausdorff semi continuity, while the lower semi continuity is implied by lower Hausdorff continuity.

Definition 1.1. A selection for the set valued map F is a map f : X [right arrow] Y such that f (x) is in F(x), for each x in X. A selection of the set valued map F which is continuous on X, is called a continuous selection ofthe map F.

It is easily verified that the metric projection onto a strongly proximinal set is upper Hausdorff metric continuous [5]. It follows from the well known Michael's selection Theorem [8, 9] that if the metric projection onto a closed, convex subset of a Banach space is lower semi continuous then it has a continuous selection.

Let X be a normed linear space. A convex, extremal subset of the closed unit ball BX is called a face of X. Let f [member of] X* and

[J.sub.X](f) = {x [member of] [S.sub.X] : f (x) = [parallel]f[parallel]}.

The functional fis called norm attaining if the set [J.sub.X](f) is non- empty. It is easily verified that if the set [J.sub.X](F) is non-empty, then it is a closed, convex and extremal subset of [B.sub.X] and is called an exposed face of [B.sub.X].

We denote by NA(X), the set of all norm attaining functionals on X and by [NA.sub.1] (X) the set NA(X) [intersection] [S.sub.X]. If H = ker f, then it is well known that [11] and [4] f [member of] NA(X) [??] [J.sub.X](f) = [empty set] [??] H is proximinal in X.

The hyperplane H is called ball proximinal if [B.sub.H] is a proximinal set. It is easily verified that ball proximinality implies proximinality.[1]

Best approximative properties of hyperplanes or subspaces in general, of Banach spaces are closely related to structure of the unit ball and its exposed faces and study of geometric structure of the unit ball in view of this link, is not new [2]. We also refer to [3], [4],[5], [7] and Proposition 1 of [10] in this regard. In [6] it was shown that if a hyperplane H, kernel of a functional f in the dual of X is ball proximinal then the set [J.sub.X](f) satisfies a restricted proximinality condition. It was also shown in that paper that exposed faces of C(Q), the Banach space of real valued continuous functions defined on a compact, Hausdorff space Q with sup norm, are proximinal sets. Here we prove that the exposed faces of the real sequence space [l.sub.1] are strongly proximinal sets and also that the metric projection onto them is Hausdorff metric continuous.

2 The Space [l.sub.1] and proximinality of exposed faces

Let N denote the set of natural numbers. We consider the real Banach space [l.sub.1], the space of sequences ([x.sub.n]) of real scalars with [parallel]x[parallel]1 = [[summation].sub.n[member of]N] |[x.sub.n]|.

In this section, we first prove that the exposed faces of the unit ball of [l.sub.1] are proximinal sets . That is, we show that if [phi] [member of] [NA.sub.1] ([l.sub.1]) then the set [mathematical expression not reproducible] is proximinal in [l.sub.1] and further characterize the set of best approximations to any element [l.sub.1], from this set.

For a real number [alpha], let

[mathematical expression not reproducible]

For a sequence x = ([x.sub.n]) of scalars, we set supp(x) = {n [member of] N : [x.sub.n] [not equal to] 0} and for any subset [LAMBDA] of N, we set

[[parallel]x[parallel].sub.[LAMBDA]] = [.summation over (n[member of][LAMBDA])] |[x.sub.n]|

Let z = ([z.sub.n]) [member of] [l.sub.[infinity]] and [[parallel]z[parallel].sub.[infinity]] = 1. If [f.sub.z] is the element of the dual of [l.sub.1], induced by z then [f.sub.z] is in [NA.sub.1]( [l.sub.1]) if and only if

{n : |[z.sub.n]| = [[parallel]z[parallel].sub.[infinity]]} = 0.

We denote the exposed face [mathematical expression not reproducible] by [mathematical expression not reproducible]. Throughout he article, the following notation is used. An element z in [l.sub.[infinity]] is fixed with [f.sub.z] in [NA.sub.1] ([l.sub.1]) and we set

[mathematical expression not reproducible]

and

[[LAMBDA].sup.+] = {n : [z.sub.n] = [[parallel]z[parallel].sub.[infinity]]}, [[LAMBDA].sup.-] = {n : [z.sub.n] = -[[parallel]z[parallel].sub.[infinity]]} and [LAMBDA] = [[LAMBDA].sup.+] [union][[LAMBDA].sup.-].

It is easily verified that

[mathematical expression not reproducible]. (1)

Also, for a fixed x in [l.sub.1], we set

[S.sup.+] = [S.sup.+.sub.x] = {n : [x.sub.n] [greater than or equal to] 0} and [S.sup.-] = [S.sup.-.sub.x] = {n : [x.sub.n] < 0}

set

[[LAMBDA].sub.1] = ([[LAMBDA].sup.+] [intersection] [S.sup.+]) [union] ([[LAMBDA].sup.-] [intersection] [S.sup.-]) and [[LAMBDA].sub.2] = ([[LAMBDA].sup.+] [intersection] [S.sup.-]) [union] ([[LAMBDA].sup.-] [intersection] [S.sup.+]). (2)

Then [LAMBDA] = [[LAMBDA].sub.1] [union] [[LAMBDA].sub.2]. Further for any y in C, it follows from (1) that [x.sub.n] and [y.sub.n] are of the same sign for n [member of] [[LAMBDA].sub.1] [intersection] supp (y) and are of opposite sign for n [member of] [[LAMBDA].sub.2] [intersection] supp (y). Thus for y in C we have

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Also for y in C we have by (2),

sgn [x.sub.n] = sgn [z.sub.n] = sgn [y.sub.n] for n [member of] [[LAMBDA].sub.1] [intersection] supp (y). (3)

In the following we prove that the set [mathematical expression not reproducible] is proximinal in [l.sub.1]. Further, for x [member of] [l.sub.1], we characterize [P.sub.C](X), the set of all best approximations to x from C. The proofs of both these results involve discussion of two cases, namely [mathematical expression not reproducible] and [mathematical expression not reproducible].

Fact 2.1. The set [mathematical expression not reproducible] is a proximinal set.

Proof. Fix x [member of] [l.sub.1]. We first assume that [[LAMBDA].sub.1] is the emptyset. Then for any y [member of] C we have [mathematical expression not reproducible] and further,

[mathematical expression not reproducible]

Hence

d(x,C) = [parallel]x[parallel] + 1 and [P.sub.C](x) = C (4)

in this case.

So we now assume that the set [[LAMBDA].sub.1] is non- empty. We now consider two cases. Case (i) [mathematical expression not reproducible].

We have

[mathematical expression not reproducible] (5)

since [mathematical expression not reproducible].

Case (ii) [mathematical expression not reproducible].

[mathematical expression not reproducible] (6)

as [mathematical expression not reproducible]. It is clear from (5) and (6) that

[mathematical expression not reproducible]. (7)

Define

[mathematical expression not reproducible].

Then [parallel]y[parallel] = 1 and using (1) and (3) we conclude y is in C. Also, [mathematical expression not reproducible]. This together with (7) implies that d(x, C) = [parallel]x - y[parallel] and therefore y is a nearest element to x from the exposed face C. This proves the proximinality of the set C. Also it is clear that

[mathematical expression not reproducible] (8)

where [mathematical expression not reproducible]

We recall that [P.sub.C](x), the set of best approximations to x from C is the set C itself, if [[LAMBDA].sub.1] is an empty set. We now characterize the set [P.sub.C](X), when the set [[LAMBDA].sub.1] is non-empty.

Fact 2.2. For x [member of] [l.sub.1] and z [member of] NA([l.sub.1]), assume that the set [[LAMBDA].sub.1] is non-empty. Then the set [P.sub.C](X), where [mathematical expression not reproducible], is given by

[mathematical expression not reproducible].

Proof. The proof involves discussion of two cases, as in the case of the previous Fact.

Case 1: [mathematical expression not reproducible].

Pick y [member of] [P.sub.C](x). Then [mathematical expression not reproducible] and also using (8), we write [mathematical expression not reproducible].

So equality holds throughout and therefore

[mathematical expression not reproducible].

Conversely if y is in C and the above two conditions are satisfied then

[mathematical expression not reproducible] and

[mathematical expression not reproducible]

from (8) and y is in [P.sub.C](X).

It is now seen if [mathematical expression not reproducible]

[mathematical expression not reproducible]. (9)

Case (ii) [mathematical expression not reproducible].

In this case, for y [member of] [P.sub.C](X), we have again using (8),

[mathematical expression not reproducible].

Hence

[mathematical expression not reproducible].

Now

[mathematical expression not reproducible].

Hence y in [P.sub.C](x) implies that [mathematical expression not reproducible].

Conversely, if y is in C we have [mathematical expression not reproducible]. Further if y is such that [mathematical expression not reproducible], then

[mathematical expression not reproducible].

Hence yis in [P.sub.C](x). Thus if [mathematical expression not reproducible]

[mathematical expression not reproducible]. (10)

3 Strong proximinality and Semi continuity of metric projection

We now prove the strong proximinality of the set [mathematical expression not reproducible].

Theorem 3.1. Let [mathematical expression not reproducible], for z in [NA.sub.1] ([l.sub.1]) and x in [l.sub.1]. Ify is in C and [parallel]x - y[parallel] < d(x, C) + [delta] for some [delta] > 0 then there exists w in [P.sub.C](X) such that [parallel]y - w[parallel] < 2[delta]. In particular, C is strongly proximinal in [l.sub.1].

Proof. If the set [[LAMBDA].sub.1], given by (2), is empty, then [P.sub.C](X) = C by (4). Hence the theorem is trivially true in this case. So we assume that the set [[LAMBDA].sub.1] is non-empty.

We have, by (8),

[mathematical expression not reproducible]

which implies

[mathematical expression not reproducible]. (11)

We now discuss three cases: [mathematical expression not reproducible]. In the following, for y [member of] C, we set

[L.sub.y] = {n [member of] [[LAMBDA].sub.1] : |[x.sub.n]| > |[y.sub.n]|}, [E.sub.y] = {n [member of] [[LAMBDA].sub.1] : |[x.sub.n]| = |[y.sub.n]|}

and

[G.sub.y] = {n [member of] [[LAMBDA].sub.1] : |[y.sub.n]| > |[x.sub.n]|}.

Then [L.sub.y] [union] [E.sub.y] [union] [G.sub.y] = [[LAMBDA].sub.1].

Case (i): [mathematical expression not reproducible].

In this case, by (11), [mathematical expression not reproducible]. Setting

[mathematical expression not reproducible].

Then [mathematical expression not reproducible]. Also, it follows from (1) that v is in C. We have [mathematical expression not reproducible] and [mathematical expression not reproducible]. So using (9), we conclude v is in [P.sub.C](x). Further

[mathematical expression not reproducible].

Taking w = v, completes the proof for this case.

Case (ii) [mathematical expression not reproducible].

Now

[mathematical expression not reproducible]

This together with (11) implies that

[mathematical expression not reproducible]

which, in turn, implies

[mathematical expression not reproducible], (12)

as [mathematical expression not reproducible].

Define v [member of] [l.sub.1] by

[mathematical expression not reproducible].

Then [parallel]y - v[parallel] < [delta] by (12) and so [parallel]v[parallel] > 1 - [delta]. Further it is clear from the definition of v that [parallel]v[parallel] [less than or equal to] [parallel]y[parallel] = 1. Thus 1 - [delta] [less than or equal to] [parallel]v[parallel] [less than or equal to] 1.

Now, using (3) we get

sgn [v.sub.n] = sgn [z.sub.n] = sgn [x.sub.n] for all n [member of] [[LAMBDA].sub.1] [intersection] supp (v). (13)

This with (1) would imply that v is in C if [parallel]v[parallel] = 1. Thus if [parallel]v[parallel] = 1, since [mathematical expression not reproducible] and

[mathematical expression not reproducible], (14)

we can conclude using (9) that v [member of] [P.sub.C](X) and take w = v to complete the proof for this case. So we assume that [parallel]v[parallel] < 1.

Let [mathematical expression not reproducible]. Then [member of] > 0 as [mathematical expression not reproducible]. Let [parallel]v[parallel] = 1 - [[delta].sub.1]. Then 0 < [[delta].sub.1] [less than or equal to] [delta]. We have |[v.sub.n]| [less than or equal to] |[x.sub.n]| for all n [member of] [[LAMBDA].sub.1] and so

[mathematical expression not reproducible].

Let [lambda] = and define

[mathematical expression not reproducible].

We have, using (13),

[mathematical expression not reproducible]

Further

[mathematical expression not reproducible]

Now [parallel]w[parallel] = 1 and also it follows from (13) that sgn [w.sub.n] = sgn [z.sub.n] for all n [member of] [[LAMBDA].sub.1]. Therefore w is in C by (1). Clearly, [mathematical expression not reproducible]. Further it is easily verified that

[mathematical expression not reproducible].

Now, using (9) and (14), we see that w is in [P.sub.C](X). Also,

[parallel]y - w[parallel]= [parallel]y - v[parallel]+ [parallel]v - w[parallel]

< [delta] + [delta] = 2[delta].

This completes the proof for this case.

Case (iii) [mathematical expression not reproducible].

Now

[mathematical expression not reproducible]

This together with (11) we have

[mathematical expression not reproducible]

Since [mathematical expression not reproducible], this in turn implies

[mathematical expression not reproducible] (15)

Define v [member of] [l.sub.1] by

[mathematical expression not reproducible].

Then, by (15), [parallel]y - v[parallel] < [delta]. Also it is clear from the definition of v that sgn [v.sub.n] = sgn [z.sub.n] for n [member of] [[LAMBDA].sub.1] [intersection] supp (v) (16)

and [parallel]v[parallel] [greater than or equal to] [parallel]y[parallel] = 1.

If [parallel]v[parallel] = 1, then v [member of] C by (1). Further, since [parallel][x.sub.n][parallel] = [parallel][y.sub.n][parallel] = [parallel][v.sub.n][parallel] for n [member of] [E.sub.y], using the definition of v we have

[mathematical expression not reproducible] (17)

Hence (10) holds and we conclude v [member of] [P.sub.C](x). We take w = v and complete the proof in this case. Therefore we assume that [parallel]v[parallel] > 1.

Now [parallel] y[parallel] = 1 and [parallel]y-v[parallel] < [delta] Therefore

1 [less than or equal to] [parallel]v[parallel] [less than or equal to] 1 + [delta].

Let [parallel]v[parallel] = 1 + [[delta].sub.1]. Then 0 < [[delta].sub.1] [less than or equal to] [delta]. Also, [mathematical expression not reproducible] and we choose [member of] > 0 such that

[mathematical expression not reproducible].

Let [lambda] = [[epsilon]/[epsilon] + [[delta].sub.1]]and define

[mathematical expression not reproducible].

Then [mathematical expression not reproducible].Moreover

[mathematical expression not reproducible]

and this implies

[mathematical expression not reproducible].

Further

[mathematical expression not reproducible]

Now [parallel]w[parallel] = 1 and using (16) we have sgn [w.sub.n] = sgn [z.sub.n] for n [member of] [[LAMBDA].sub.1] [intersection] supp (w).

This with (1) implies w [member of] C.

It is easily verified using (17) that

[mathematical expression not reproducible]

and using (10), we conclude that w [member of] [P.sub.C](x). Finally

[parallel]y - w[parallel] = [parallel]y - v[parallel] + [parallel]v - w[parallel]

< [delta] + [delta] = 2[delta].

and this completes the proof of the theorem.

We now proceed to show that the metric projection [P.sub.C] is Hausdorff metric continuous. Since C is strongly proximinal the metric pro ection [P.sub.C] is upper Hausdorff semi-continuous and we need to prove only the lower Hausdorff semi-continuity of [P.sub.C].We need the following Fact for this purpose.

Fact 3.2. Let C be a closed convex subset of a Banach space X, which satisfies a uniform version of strong proximinality, namely: there exists a function [phi] from [0, + [infinity]) to itself such that [phi](0) = 0 and [phi] is right-continuous at 0, such that for any x [member of] X and any [delta] > 0, if y [member of] C satisfies [parallel]x - y[parallel] < d(x, C) + [delta], there exist w [member of] [P.sub.C]( x) such that [parallel]y - w[parallel] [less than or equal to] [phi]([delta]). Then the metric projection [P.sub.C] is Hausdorff metric continuous.

Proof. Let x [member of] [l.sub.1] and [epsilon] > 0 be given. We can choose [delta] > 0, such that [phi]([delta]) < [epsilon], since [phi](0) = 0 and [phi] is right-continuous at 0. We now claim that for any y [member of] [P.sub.C](X) and z [member of] [l.sub.1] with [parallel]x - z[parallel] < [[delta]/2], there exists w [member of] [P.sub.C](Z) satisfying [parallel]y - w[parallel] < [epsilon]. This would clearly imply the lower Hausdorff semi-continuity of the set valued map [P.sub.C] at x.

Since [parallel]x - z[parallel] < [[delta]/2], we have

|d(x, C) - d(z, C)| [less than or equal to] [parallel]x - z[parallel] < - [[delta]/2].

Thus

[mathematical expression not reproducible]

Now, using our assumption, we conclude there exists w [member of] [P.sub.C](z) such that [parallel]y - w[parallel] < [phi]([delta]) < [epsilon].

The above Fact in conjunction with Theorem 3.1 implies Hausdorff metric continuity of the map [P.sub.C], as given below.

Fact 3.3. If z in [NA.sub.1] and [mathematical expression not reproducible], then the metric projection [P.sub.C] is Hausdorff metric continuous.

Proof. It follows from the statement of Theorem 3.1 that [phi]([delta]) = 2[delta] in this case. The map [phi] clearly satisfies the conditions of Fact 3.2 and hence it follows that the metric projection [P.sub.C] is Hausdorff metric continuous.

Thus [P.sub.C] is Hausdorff metric continuous and by the well known Michael selection theorem [8, 9], [P.sub.C] has a continuous selection.

Acknowledgements

The author would like to thank Professor V.Indumathi for many helpful discussions and valuable suggestions throughout this work. She would also like to express her thanks to the Referee for his useful comments and suggestions.

References

[1] P. Bandyopadhyay, Bor-Luh Lin and T.S.S.R.K. Rao, Ball proximinality in Banach Spaces, in: Banach Spaces and Their Applications in Analysis(Oxford/USA,2006)B.Randrianantoanina et all (ed.), Proceedings in Mathematics, de Gruyter, Berlin (2007) 251-264.

[2] Frank Deutsch and Robert J. Lindahl, Minimal Extremal Subsets of the Unit Sphere, Mathematish Annalen, 197 (1972) 251-278.

[3] Gilles Godefroy, The Sequence Space c0, Extracta Mathematicae, Vol.16, No.1, 2001)1-25.

[4] Gilles Godefroy and V. Indumathi, Proximinality in subspaces of c0, J.Approx. Th, 101(1999)175-181.

[5] Gilles Godefroy and V. Indumathi, Strong proximinality and polyhedral spaces, Rev. Mat. complut. 14 (2001) 105-125.

[6] V. Indumathi and S. Lalithambigai, Ball Proximinal Spaces, Journal of Convex Analysis Vol.18, No.2,(2011)353-366.

[7] J.F. Mena, R. Paya, A. Rodriguez and D.T. Yost, Absolutely Proximinal Subspaces of Banach Spaces Journal of Approximation theory, Vol.65, No.1 (1991)46-72.

[8] E. Michael, Continuous selections, I, Ann.Math. 63(1956)361-382.

[9] E. Michael, Selected selection theorems, Amer. Math. Monthly, 63(1956),233-238.

[10] R. Paya and D. Yost, The two Ball Property:Transitivity and Examples Mathematika, 35(1988)190-197.

[11] Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, 1970.

Department of Mathematics

Pondicherry University

Pondicherry-605014, India

Email: kalaiarasi.math@gmail.com

Received by the editors in July 2016 - In revised form in October 2016.

Communicated by G. Godefroy.

Key words and phrases : Proximinal, strongly proximinal, norm attaining functional, exposed face, Hausdorff metric continuity.
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