ON PROJECTION CONSTANT PROBLEMS AND THE EXISTENCE OF METRIC PROJECTIONS

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ON PROJECTION CONSTANT PROBLEMS AND THE EXISTENCE OF METRIC PROJECTIONS

IN NORMED SPACES

E. M. EL-SHOBAKY, SAHAR MOHAMMED ALI, AND WATARU TAKAHASHI

Received 3 September 2001

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spaces lp, 1≤p <∞andc₀. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator fromlp, 1≤p <∞orc₀ onto anyone of their maximal proper subspaces.

1. Introduction

A subsetCof a normed linear spaceXis called an existence subset ofXif and only if for every elementx∈Xthere is an elementy∈C, whereyis called the best approximation ofxdenoted byb(x, C)[2,3] such that

x−y =dist(x, C):=inf

x−y :y∈C

. (1.1)

Needless to say, best approximations play a major role in many applications, including approximation theory, optimization and applications in mathematical economics and engineering. Thus, the mathematical analysis of the properties of the best approximation elements has drawn much attention in research.

In [11,13], the authors there used the terms ˇCebyšev subsets and proximal subsets instead of the existence subsets and studied the characterizations of the existence subsets of Banach spaces.

It is shown that ifXis a reflexive Banach space and Cis a closed convex subset, then for everyx∈Xthe best approximation elementb(x, C)exists and is unique.

2000 Mathematics Subject Classification: 41A50, 41A52, 46A32, 46N10 URL:http://aaa.hindawi.com/volume-6/S1085337501000732.html

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In this paper, reduced assumptions on a normed linear space for a closed convex subset to exist are given, instead of the reflexivity and the completeness assumptions of the given normed space.

On the other hand, it is known that the existence of a projectionP from a Banach spaceXonto its closed subspaceY is equivalent to the existence of an extension T of any operatorT fromY intoW to an operator fromXintoW such thatT ≤ PT. The two equivalent problems, that is

(1) how small can the norm of the extended operator be made? and (2) what is the projection of smallest norm?,

are challenging to the study of the relative projection constantλ(Y, X)ofY in X, where

λ(Y, X):=inf

P, P is a projection fromXontoY

(1.2) and the absolute projection constant ofY,λ(Y ), where

λ(Y ):=sup

λ(Y, X), XcontainsY as a closed subspace

. (1.3) In [10], the upper estimate for the absolute projection constantλ(Y )of a finite- dimensional spaceY with dimY=nis found in the form

λ(Y )≤









√n− 1

√n+O

n^−3/4 , in the real field,

√n− 1 2√

n+O

n^−3/4 , in the complex field.

(1.4)

The precise values forl₁ⁿ, l₂ⁿ, andln^p, p= 1, p=2, have been calculated by Grünbaum [8], Gordon [7], Garling and Gordon [6], Rutovitz [12], and König et al. [9].

In [5], interesting results have been given for the injective and projective tensor products.

For a finite codimensional subspaces, Garling and Gordon [6] showed that if Y is a finite codimensional subspace of the spaceXwith co-dimensionn, then for every >0 there exists a projectionP fromXontoY with norm

P ≤1+(1+)√

n. (1.5)

And so,λ(Y, X)≤1+√

n. In particular, if the co-dimension ofY is 1, that is, Y is a hyperplane in the spaceX, thenλ(Y, X)≤2.

2. Notation and basic definitions

We will use the same notation that is given in [1,4,14].

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LetCbe a nonempty closed convex subset of a normed spaceX. If for every x∈Xthere is a uniqueb(x, C)inC, then the mappingb(x, C)is said to be a metric projection ontoC, in this case we have

x−b(x, C)=dist(x, C) ∀x∈X. (2.1) Clearly, ifXis a Hilbert space andCis a nonempty closed convex subset ofX, then there is a metric projection fromXontoC, see [14].

As a direct consequence of the separation theorem, ifXis a locally convex linear topological space, then a nonempty convex subsetC ofX is closed in the strong topology ofXif and only ifCis closed in the weak topology ofX, see [4].

A relative projection constant of the closed subspaceYin the spaceXis said to be exact if and only if there is a projectionP fromXontoY at which the infimum of (1.2) is attained.

A subspaceYof the spaceXis said to be a hyperplane (maximal proper subspace) of the spaceXif and only ifXcontainsY as a subspace of deficiency 1.

It is known that a subspaceY is a hyperplane of the spaceXif and only if there is a functionalf ∈X^∗such thatY =f⁻¹({0}); see [1].

LetP be an operator on the spaceX. Then the pointx∈Xis said to be a maximal point of the operatorP if and only ifP = P (x).

IfXis either the Banach spacel_∞, the Banach space of all bounded scalar- valued functions{xn}^∞_n₌₁on a countably infinite setN, orlp the Banach space of all scalar-valued functionsx= {xn}^∞n=1 on a countably infinite setN, such that_∞

n=1|xn|^p<∞orc₀ the closed subspace of the Banach spacel_∞, then the norms onXare defined as follows:

xX:=











sup^∞_n₌₁xn ifX=l_∞, _∞

n=1

xn^p1/p

ifX=lp.

(2.2)

Our first result is the following theorem.

Theorem2.1. LetXbe a normed space in which every Cauchy sequence has a weakly convergent subsequence and the parallelogram law holds. LetCbe a nonempty closed convex subset ofX. Then the metric projectionb(·, C)fromX ontoCexists.

Proof. Let x∈X and consider the distance function dist(x, C), there exists a sequence{yn}^∞_n₌₁of elements inCsuch that

dist(x, C)≤x−yn≤dist(x, C)+1

n, n=1,2, . . . . (2.3)

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Taking the limit asn→ ∞, we get limn→∞x−yn =dist(x, C). The sequence {yn}^∞_n₌₁is a Cauchy sequence inX. In fact, using the convexity ofC, we have (1/2)(yi+yj)∈C, also using the parallelogram law, we have

yi−x −

yj−x ²+yi−x +

yj−x ²=2yi−x²+2yj−x². (2.4) Therefore,

yi−yj²=2yi−x²+2yj−x²−4 x−1

2

yi+yj

2

≤2

dist(x, C)+1 i

2

+2

dist(x, C)+1 j

2

−4 dist(x, C)²−−−−→^i,j^→∞ 0,

(2.5)

using the assumption, every Cauchy sequence has a weakly convergent subsequence, the sequence{yn}^∞_n=1 has a subsequence{yni}^∞_i=1 converging weakly to some point y₀ in X, yn_i

i→ ∞

−−−→weakly y₀. Since C is convex and is closed in the strong topology,Cis closed in the weak topology, thenCcontains as well all its weak limitsy0∈C. Now, consider the proper convex lower semi-continuous real-valued functiongonCdefined by

g(z)= x−z ∀z∈C, (2.6)

we have

nlim→∞g

yn = lim

n→∞x−yn=dist(x, C), (2.7) and the mappinggattains its minimum aty0. In fact, let >0. Then the set

Gg(y₀)−=

z∈C:g(z)≤g y₀ −

(2.8) is a convex closed subset ofX. Using the convexity ofGg(y0)−, for every >0 the setGg(y0)− is a weakly closed subset ofCand hence for every >0 the set

G^c(C)_g(y

0)−=

z∈C:g(z) > g y₀ −

(2.9) is a weakly open subset ofC, sincey0∈G^c(C)_g(y

0)−, there is a neighborhoodV ofy₀(w.r.t. the weak topology) such thatV ⊂G^c(C)_g(y

0)−. Using the weak limit point definition of the sequence{yn_i}^∞_i₌₁, there isi0∈Nsuch thatyn_i ∈V for alli≥i₀, thenyni ∈G^c(C)_g(y

0)− for alli≥i₀. Thereforeg(yni) > g(y₀)−for everyi≥i0. Finally we have

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inf

g(z):z∈C

≤g

y0 ≤inf

i≥i0

g

yni +≤lim inf

i g yni +

= lim

i→∞g

yn_i += lim

n→∞g yn +

=inf

g(z):z∈C +.

(2.10)

Since >0 is arbitrary andy0∈C, we have g

y₀ =min

g(z):z∈C

. (2.11)

Thus for everyx∈Xthere isy₀∈Csuch that x−y₀=min

x−z :z∈C

=dist(x, C). (2.12) To show that such a point y₀ is unique, let g(y₀)=0. Sincex−y₀ =0, x=y₀,y₀is unique. Letg(y₀) >0 and letz₀be an element inCwith

g

z0 =x−z0=min

x−z :z∈C

=g

y0 =x−y0 (2.13) andz0=y0(z0−y0>0), sinceCis convex,(1/2)(y₀+z0)∈C, using the parallelogram law, we have

g y0 ≤g

1 2

y0+z0 = x−1

2

y0+z0

=1

2x−y₀ +

x−z₀ =1 2

x−y₀ +

x−z₀ ² _1/2

=1 2

2x−y0 2

+2x−z0 2

−z0−y0²1/2

<1 2

2x−y0 2

+2x−z0 21/2

=1 2

2g

y₀ ²+2g z₀ ²

1/2

=g y₀ .

(2.14)

This is a contradiction. Therefore no suchz₀exists, andy₀is unique. Now, define the mappingb(x, C)fromXontoCbyb(x, C)=y0, the mappingb(x, C)is

the required metric projection.

Our result concerning the maximal proper subspaces of the classical Banach spacesc0orlp for 1≤p <∞is the following result.

Theorem2.2. LetXdenote one of the spacesc₀orlpfor1≤p <∞,f ∈X^∗ and Y the closed linear subspace Y = f⁻¹({0}) = {y = {bi}^∞_i=1 : f (y) = _∞

i=1bifi =0} of the space X. Then the general formula of any projection fromXontoY is given by

P=Ilp−f⊗z for somez∈Xwithf (z)=1. (2.15)

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Proof. Since every element x = {xi}^∞_i₌₁ ∈ X is uniquely written as x = _∞

i=1xiei, any operatorP onXis completely determined byP (ei), suppose thatP (ei)= {eik}^∞_k₌₁, we have

P (x)=

i≥1

xiP

ei =

i≥1

xieik

∞ k=1

. (2.16)

Define the elementz= {αk}^∞_k₌₁∈Xas follows

fiαk=δik−eik, k≥1. (2.17) Sincef is a nonzero element of the spacel₁orlq, where 1/p+1/q=1,f =0, there is at least one indexifor whichfi=0, for this index multiplying (2.17) byfk and summing with respect tok, we get

fi

k≥1

fkαk=

k≥1

fk

δik−eik =fi−

k≥1

fkeik. (2.18)

Since P (ei) ∈ Y and f (P (ei)) = 0, that is,

k≥1fkeik = 0. Therefore, fi

k≥1fkαk =fi, this proves that

k≥1fkαk = 1, that is, f (z) =1. On the other hand, we have eik =δik−αkfi. Thus the representation ofP is as follows:

P (x)=





i≥1

xi

δik−αkfi





∞

k=1

=



xk−αk





i≥

xifi









∞

k=1

=

xk−αkf (x)_∞

k=1=x−f (x) αk

_∞

k=1=x−f (x)z.

(2.19)

The converse direction is clearly true. To calculate the norm of the given projec- tionP, we have two distinct cases, the first is whenX=c₀in which the norm is as follows:

P = sup

x=1



 ∞ i=1

δikxi−αk

i≥1

xifi





n

k=1

l_∞

=sup

k≥1

1−αkfk+αkfl₁−fk .

(2.20)

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The second case is whenX=lp in which the norm ofP is as follows:

P = sup

x=1





i≥1

δikai−αk

i≥1

xifi





∞

k=1

l_p

= sup

x=1





i≥1

δik−αkfi xi





∞

k=1

l_p

= sup

x=1

δik−αkfi

_∞

i=1

xi

_∞

i=1 ∞ k=1

l_p

= sup

x=1





k≥1

δik−αkfi

_∞

i=1

xi

_∞

i=1 p





1/p

=





k≥1

sup

x=1

δik−αkfi

_∞

i=1

xi

_∞

i=1 p





1/p

=





k≥1

δik−αkfi

_∞

i=1

lq

p





1/p

=





k≥1

_∞

i=1

δik−αkfi^qp/q



1/p

. (2.21) Corollary2.3. Iff = {fi}^∞_i₌₁ is an element of the space (l₁)^∗=l_∞ andY is a subspace of the spacel₁,Y =f⁻¹({0}), then some of the maximal points of any projectionP froml₁ ontoY lie in the set {ei}^∞_i₌₁, where{ei}^∞_i₌₁ is the canonical basis ofl1.

Proof. According to Theorem 2.2, the norm of any projection in this case is given by

P = sup

x=1

p(x)= sup

x=1

x−f (x)z₀

= sup

x=1





i≥1

δikxi−αk

i≥1

xifi





∞

k=1

l₁

≤

k≥1 x=1sup

δik−αkfi

_∞

i=1

xi

_∞

i=1

=

k≥1

δik−αkfi

_∞

i=1

l_∞=

k≥1

sup

i≥1

δik−αkfi

=sup

i≥1

k≥1

δik−αkfi=sup

i≥1

P ei .

(2.22)

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Therefore the norm of any such projection is attained at some points in the set

{ei}^∞_i₌₁.

Remark 2.4. If one of the coordinates off, sayfi=0, thenei∈Y(forf (ei)= fi=0). And to get a norm one projection the effect of this projection on the basis elements must not exceed one.

Corollary2.5. Let Xdenote one of the spaces c0 orlp,1< p <∞, and let f ∈X^∗. Then an infinite matrix P = {pin}i,n∈N represents a projection operator from X onto its hyperplanef⁻¹({0})if and only if the matrix α= {δin−pin}i,n∈Nsatisfies the following conditions:

(1)traceα=1,

(2)each row of the matrixαis a scalar multiple off.

Proof. Let P = {pin}i,n∈N be an infinite matrix satisfying conditions (1) and (2). ThenP =I−α, since each row of the matrixαis a scalar multiple off, α(y)=0 for ally∈f⁻¹({0}). And so,P (y)=yfor all y∈f⁻¹({0}). Iffⁿ denotes thenth row of the matrixα, using condition (2) we obtain for eachn∈N a scalarαnsuch thatfⁿ=αnf, using condition (1) we havef (z)=1, where z= {αn}^∞_n₌₁. To show that the range ofP isf⁻¹({0})it is sufficient to show thatf (P (x))=0 for allx= {xi}^∞_i₌₁∈X, so suppose thatx = {xi}^∞_i₌₁∈X, we have

f

P (x) =f (I−α)(x)=f (x)−f

α(x) =f (x)−f αnfi

i,n∈N(x)

=f (x)−f





!_∞

i=1

αnfixi

"

n∈N



=f (x)− ∞ n=1

fn

∞ i=1

αnfixi

=f (x)− ∞ n=1

fnαn× ∞ i=1

fixi=f (x)−f (z)f (x)=0.

(2.23) Conversely, ifP is a projection fromXontof⁻¹({0}), usingTheorem 2.2, we obtain an elementz= {αn}^∞_n₌₁∈Xsuch thatf (z)=1 andP=IX−f⊗z. So

P xi

_∞

i=1 =IX(x)−f xi

_∞

i=1 ×z=IX(x)− ∞ i=1

fixi×z

=IX(x)−

! zn

∞ i=1

fixi

"_∞

n=1

=

δin−αin

i,n∈N

xi

_∞

i=1 , (2.24)

whereαin=zifn. This proves that the operatorP has the matrix representation P= {δin−αin}i,n∈N. Clearly the matrixα= {αin}i,n∈Nsatisfies conditions (1)

and (2).

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Corollary2.6. Letf =δ{k}ⁿ_k₌₁,n >2, be a sequence of the spacel_n¹, where δis a nonzero scalar andi= ±1. Then the relative projection constant of the (n−1)-dimensional subspacef⁻¹({0})in the spacel_n^∞is given by

λ f⁻¹

{0} , l_n^∞ =2−2

n. (2.25)

Moreover, the minimal norm projection is given by P₀(x)=x− f (x)

f²_l2 n

×f. (2.26)

Proof. As given inTheorem 2.2the norm of any projection corresponding to the elementz= {αk}ⁿ_k=1is given by

P =supⁿ

k=1

1−αkfk+αkfl¹_n−fk

=supⁿ

k=1

1−δαkk+αk[n−1]|δ| .

(2.27)

Assume that the minimal projection is a norm one projection. Then there is z∈l¹_nand

1−δαkk+[n−1]αk|δ| ≤1, (2.28) for everyk=1,2, . . . , n. In this case, we have 1−|δαk|+[n−1]|αk||δ| ≤1 for everyk=1,2, . . . , n. Therefore[n−2]|δ|αk| ≤0 for everyk=1,2, . . . , n. For n >2, this is true only ifαk=0 for everyk=1,2, . . . , n. This is an obvious contradiction, thus there is no norm one projection froml_n¹ontof⁻¹({0}).

Now, letx= {xk}ⁿ_k₌₁∈l_n^∞be an arbitrary point. To project this point to the point x₀ = {x_k⁰}ⁿ_k=1 in the space f⁻¹({0}) with a minimal available distance between the points x = {xk}ⁿ_k₌₁ and x₀ = {x_k⁰}ⁿ_k₌₁, the sequence x₀−x = {x_k⁰−xk}ⁿk=1 must be parallel to the line passing through f = {fk}ⁿk=1 and perpendicular to the planef⁻¹({0}). Thus there is a scalarλsuch thatx₀−x= λf. On the other hand, since x0 ∈f⁻¹({0}), f (x0)= 0, thus 0=f (x0)= f (x)+λf²_l2

nand soλ= −f (x)/f²_l2

n, it follows thatx₀=x−f (x)/f²_l2 n× f. The required projectionP₀froml_n^∞ontof⁻¹({0})is defined by the formula

P₀ x=

xk

n

k=1 =x₀=x− f (x) f²_l2 n

×f. (2.29)

(Note that the elementz₀corresponding toP₀isz₀=f/f²_l2

nand alsoP₀ = (2−2/n).)

Now we are going to show that this projection is a minimal norm projection.

Assume the contrary, that is, there exists an elementz∈l_n¹such thatf (z)=1

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and the corresponding projection P satisfiesP< (2−2/n), according to (2.27), we have

1−δαkk+αk[n−1]|δ| <

2−2

n

(2.30) for everyk∈ {1,2, . . . , n}, from which we get

[n−2]αk|δ|<

1−2

n

(2.31) and so for suchzwe have

αk< 1

n|δ| ∀k=1,2, . . . , n, (2.32) multiplying by|fk| = |δ|, summing with respect tok, we getn

k=1|fk||αk|<1.

On the other hand, the inequality 1=

n k=1

fkαk≤ n k=1

fkαk<1 (2.33) gives a contradiction, hence no suchzexists, from which we concluded the proof.

Corollary2.7. Iff = {fn}^∞n=1 is a sequence of the spacel1, andf⁻¹({0}).

Then λ(f⁻¹({0}), c₀) = 1 if and only if there is n ∈ N for which |fn| ≥ (1/2)fl1.

Proof. As given inTheorem 2.2, the norm of any projection corresponding to the elementzis given by

P =sup

k≥1

1−αkfk+αkfl1−fk . (2.34) To have a norm one projection, we must have|1−αkfk|+|αk|(fl1−|fk|)≤1 for everyk∈N. In this case we have 1− |αkfk| + |αk|(fl1− |fk|)≤1 for everyk∈N. Therefore

αkfl1−2fk ≤0 ∀k∈N. (2.35) Ifk∈N and(fl1−2|fk|) >0, then|αk| =0, butf (z₀)=1 implies that at least onek∈Nfor which|αk| =0 and so at least onek∈Nfor which(f^l1− 2|fk|)≤0. Therefore there is at leastk∈Nfor which|fk| ≥(1/2)fl₁. Example2.8. The minimal norm projection of the subspaceY= {x= {bi}³i=1| ₃

i=1bi=0}in the spacel₃^∞is the projection given by P₀

xi

3 i=1

=1 3

2x₁−x₂−x₃,2x₂−x₁−x₃,2x₃−x₁−x₂

. (2.36) with normP₀ =4/3.

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References

[1] M. M. Day,Normed Linear Spaces, Ergebnisse der Mathematik und ihrer Gren- zgebiete. Neue Folge. Heft, vol. 21, Springer-Verlag, Berlin, 1958.MR 20:1187.

Zbl 0082.10603.

[2] F. S. De Blasi and J. Myjak, On a generalized best approximation problem, J.

Approx. Theory94(1998), no. 1, 54–72.MR 99i:41034. Zbl 0912.41015.

[3] F. S. De Blasi, J. Myjak, and P. L. Papini, Porous sets in best approximation theory, J. London Math. Soc. (2)44 (1991), no. 1, 135–142.MR 92h:41066.

Zbl 0786.41027.

[4] N. Dunford and J. T. Schwartz,Linear Operators. I. General Theory, Pure and Applied Mathematics, vol. 6, Interscience, New York, 1958. MR 22:8302.

Zbl 0084.10402.

[5] E. M. El-Shobaky, S. M. Ali, and W. Takahashi,On the projection constants of some topological spaces and some applications, Abstr. Appl. Anal.6(2001), no. 5, 299–308.

[6] D. J. H. Garling and Y. Gordon,Relations between some constants associated with finite dimensional Banach spaces, Israel J. Math.9(1971), 346–361.MR 54:896.

Zbl 0212.14203.

[7] Y. Gordon,On the projection and Macphail constants ofl_pⁿspaces, Israel J. Math.

6(1968), 295–302.MR 38:4961. Zbl 0182.45202.

[8] B. Grünbaum,Projection constants, Trans. Amer. Math. Soc.95(1960), 451–465.

MR 22:4937. Zbl 0095.09002.

[9] H. König, C. Schütt, and N. Tomczak-Jaegermann,Projection constants of symmet- ric spaces and variants of Khintchine’s inequality, J. Reine Angew. Math.511 (1999), 1–42.MR 2000i:46014. Zbl 0926.46008.

[10] H. König and N. Tomczak-Jaegermann,Norms of minimal projections, J. Funct.

Anal.119(1994), no. 2, 253–280.MR 94m:46024. Zbl 0818.46015.

[11] R. R. Phelps,Cebyšev subspaces of finite codimension inˇ C(X), Pacific J. Math.13 (1963), 647–655.MR 27:6069. Zbl 0115.10101.

[12] D. Rutovitz,Some parameters associated with finite-dimensional Banach spaces, J.

London Math. Soc.40(1965), 241–255.MR 32:8120. Zbl 0125.06402.

[13] I. Singer,Best Approximation in Normed Linear Spaces by Elements of Linear Sub- spaces, Die Grundlehren der mathematischen Wissenschaften, vol. 171, Springer- Verlag, New York, 1970.MR 42:4937. Zbl 0197.38601.

[14] W. Takahashi,Nonlinear Functional Analysis, Fixed Point Theory and Its Applica- tions, Yokohama Publishers, Yokohama, 2000.

E. M. El-Shobaky: Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

E-mail address:[email protected]

Sahar Mohammed Ali: Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

Wataru Takahashi: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology,2-12-1Ookayama, Meguro-ku, Tokyo152-8552, Japan