# New Generalization off-Best Simultaneous Approximation in Topological Vector Spaces

Authors: Mahmoud Rawashdeh; Sarah Khalil
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

LetK be a nonempty subset of a Hausdorff topological vector space X, and let f be a real-valued continuous function on X. If for each x=(x1,x2,…,xn)∈Xn, there exists k0∈K such that FK(x)=∑i=1nfxi-k0=inf∑i=1nf(xi-k):k∈K, then K is called f-simultaneously proximal and k0 is called f-best simultaneous approximation for x in K. In this paper, we study the problem of f-simultaneous approximation for a vector subspace K in X. Some other results regarding f-simultaneous approximation in quotient space are presented.


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## Body

## 1. Introduction

LetK be a closed subset of a Hausdorff topological vector space X and f a real-valued continuous function on X. For x∈X, set FK(x)=infk∈Kf(x-k). A point k0∈K is called f-best approximation to x in K if FK(x)=f(x-k0). The set PKf(x)={k∈K:FK(x)=f(x-κ)} denotes the set of all f-best approximations to x in K. Note that this set may be empty. The set K is said to be f-proximal (f-Chebyshev) if for each x∈X, PKf(x) is nonempty (singleton). The notion of f-best approximation in a vector space X was given by Breckner and Brosowski [1] and in a Hausdorff topological space X by Narang [2, 3]. For a Hausdorff locally convex topological vector space and a continuous sublinear functional f on X, certain results on best approximation relative to the functional f were proved in [1, 4]. By using the existence of elements of f-best approximation, certain results on fixed points were proved by Pai and Veermani in [5]. In addition, for a topological vector space X relative to upper semicontinuous functions, some results on best approximation were proved by Haddadi and Hamzenejad [6]. Moreover, Naidu [7] proved some results on best simultaneous approximation related to f-nearest point and topological vector space X.Analogous to the problem of simultaneous approximation [8], we introduce the concept of best f-simultaneous approximation as follows.Definition 1.
LetK be a non-empty subset of a Hausdorff topological vector space X, and let f be a real-valued continuous function on X. A point k0∈K is called f-best simultaneous approximation in K if there exists x=(x1,x2,…,xn)∈Xn such that
(1)FK(x)=inf{∑‍ni=1f(xi-k):k∈K}=∑‍i=1nf(xi-k0).The set of allf-best simultaneous approximations to x=(x1,x2,…,xn)∈Xn in K is denoted by
(2)PKf(x)={k∈K:FK(x)=∑‍ni=1f(xi-k)}.The setK is called f-simultaneously proximal (f-simultaneously Chebyshev) if for each x=(x1,x2,…,xn)∈Xn, PKf(x)≠ϕ (singleton). If n=1, simultaneous f-proximal is precisely f-proximal.We remark that iff(x)=∥x∥, then the concept of f-best approximation is precisely the best approximation.A setK is said to be inf-compact at a point x=(x1,x2,…,xn)∈Xn [5] if each minimizing sequence in K (i.e., ∑i=1nf(xi-kn)→FK(x)) has a convergent subsequence in K. The set K is called inf-compact if it is inf-compact at each x=(x1,x2,…,xn)∈Xn.It is easy to see that ifK is compact or inf-compact, then K is f-simultaneously proximal.In this paper, we introduce the concept off-simultaneous approximation and study the existence and uniqueness problem of f-simultaneous approximation of a subspace K of a Hausdorff topological vector space X. Certain results regarding f-simultaneous approximation in quotient spaces are obtained by generalizing some of the results in [9].Throughout this paper,X is a Hausdorff topological vector space and f is a real-valued continuous function on X.


## 2.f-Simultaneous Approximation

In this section, we give some characterizations off-proximal sets in X. We begin with the following definitions.Definition 2.
A functionf:X→ℝ is called absolutely homogeneous if f(αx)=|α|f(x), for all x∈X and all α∈ℝ.Definition 3.
A subsetK of X is called f-closed if for all sequences {km} of K and for all x=(x1,x2,…,xn)∈Xn, such that ∑i=1nf(xi-km)→0, we have x∈Kn.Definition 4.
A subsetK of X is called f-compact if for every sequence {kn} in K there exist a subsequence {knk} of {kn} and k0∈K such that f(knk-k0)→0.Definition 5.
Forx,y∈X, where x=(x1,x2,…,xn)∈Xn and y=(y1,y2,…,yn)∈Xn, x is said to be f-orthogonal to y denoted by x⊥fy, if ∑i=1nf(xi)≤∑i=1nf(xi+αyi) for every scalar α∈ℝ. Also, x is said to be f-orthogonal to a set K if x⊥fk, for all k∈K.Definition 6.
We say thatK is w-compact if every net {kα} in K has a convergent subnet.Theorem 7.
LetK be a subset of X. Then, one has the following.(1)
FK+y(x+Y)=FK(x), for all x=(x1,x2,…,xn), where Y=(y,y,…,y)∈Xn.(2)
PK+yf(x+Y)=PKf(x)+y, for all x=(x1,x2,…,xn).(3)
K is f-simultaneously proximal (f-simultaneously Chebyshev) if and only if K+y is f-simultaneously proximal (f-simultaneously Chebyshev) for every y∈X.Moreover, iff is absolutely homogeneous function, then one has the following.(4)
FαK(αx)=|α|FK(x), for all x=(x1,x2,…,xn)∈Xn and α∈ℝ.(5)
PαKf(αx)=αPKF(x), for all x=(x1,x2,…,xn)∈Xn and α∈ℝ.(6)
K is f-simultaneously proximal (f-simultaneously Chebyshev) if and only if αK is f-simultaneously proximal (f-simultaneously Chebyshev), α∈ℝ.(7)
Iff is convex function and K is a convex set, then PKf(x) is convex.Proof.
(
1
) Let x=(x1,x2,…,xn) and Y=(y,y,…,y)∈Xn. Then
(3)FK+y(x+Y)=infk∈K∑‍i=1nf((xi+y)-(κ+y))=FK(x).
(
2
) The equation
(4)∑i=1nf(xi-k0)=infk∈K∑‍i=1nf((xi+y)-(k+y))=infk∈K∑‍i=1nf(xi-k)
implies that k0+y∈PK+yf(x+Y) if and only if k0∈PKf(x). Thus,
(5)PK+yf(x+Y)=PKf(x)+y.
(
3
) The proof follows immediately from part (2) above.
(
4
) Let x=(x1,x2,…,xn)∈Xn, α∈ℝ. Then,
(6)FαK(αx)=infk∈K∑‍i=1nf(αxi-αk)=|α|infk∈K∑‍i=1nf(xi-k)=|α|FK(x).
(
5
) If α=0, then we are done. If α≠0 and k0∈PαKf(αx), then k0∈αK and
(7)∑‍i=1nf(αxi-k0)=infk∈K∑‍i=1nf(αxi-αk).
This implies that
(8)∑‍i=1nf(xi-1αk0)=FK(x),
which implies that (1/α)k0∈PKf(x).
(
6
) The proof follows immediately from part (5) above.
(
7
) Let k1, k2∈PKf(x). Since K is convex, then λk1-(1-λ)k2∈K. We must show that λk1-(1-λ)k2∈PKf(x); that is,
(9)∑‍i=1nf(xi-(λk1-(1-λ)k2))=infk∈K∑‍i=1nf(xi-k).
So,
(10)∑‍i=1nf(xi-(λk1-(1-λ)k2))=∑‍i=1nf(λ(xi-k1)+(1-λ)(xi-k2))=λ∑‍i=1nf(xi-k1)+(1-λ)∑‍i=1nf(xi-k2)=λFK(x)+(1-λ)FK(x)=FK(x)=∑‍i=1nf(xi-k),
which implies that PKf(x) is convex.Example 8.
LetX=ℝ2 and K={(x1,x2)∈ℝ2:x12+x22≤4}, and let f(x,y)=x2-y2. If z=((0,0),(0,1))∈X2, then one can show that FK(z)=f(0,1/2)=-1/4.Theorem 9.
Letf be an absolutely homogeneous real-valued function on X and M a vector subspace of X. Then,(1)
FM(αx)=|α|FM(x), for all x=(x1,x2,…,xn)∈Xn, α∈ℝ-{0};(2)
PMf(αx)=αPMf(x), for all x=(x1,x2,…,xn)∈Xn, α∈ℝ-{0}.Proof.
(
1
) Let x=(x1,x2,…,xn). Then,
(11)FM(αx)=infm∈M∑‍i=1nf(αxi-m)=|α|infm′∈M∑‍i=1nf(xi-m′)=|α|FM(x).
(
2
) Let m0∈PMf(αx). Then,
(12)∑‍i=1nf(αxi-m0)=infm∈M∑‍i=1nf(αxi-m)
if and only if
(13)∑‍i=1nf(xi-1αm0)=infm′∈M∑‍nf(xi-m′)=FM(x),
for all α∈ℝ-{0}, which implies that (1/α)m0∈PMf(x), so, m0∈αPMf(x).Theorem 10.
Letf be a positive real-valued function on X such that x=0 if and only if f(x)=0. Then, if K is f-simultaneously proximal, then K is f-closed.Proof .
Sincef is a positive function, then ∑i=1nf(xi)≥0 for all x=(x1,x2,…,xn)∈Xn. Let {km} be a sequence of K and x=(x1,x2,…,xn)∈Xn, such that ∑i=1nf(xi-km)→0. This implies that
(14)FK(x)=infk∈K∑‍i=1nf(xi-k)≤∑‍i=1nf(xi-km)⟶0.
Since K is f-simultaneously proximal, then there exists k0∈K such that
(15)FK(x)=∑‍i=1nf(xi-k0)=0.
Hence, for alli=1,2,…,n, f(xi-k0)=0.Using the assumption it follows that xi-k0=0, and, hence, xi=k0∈K. Consequently, x∈Kn and K is f-closed.Theorem 11.
LetX be a topological vector space and K a vector subspace of X. Suppose that f is continuous function and K is w-compact; then, K is f-simultaneously proximal.Proof.
Letx=(x1,x2,…,xn)∈Xn. Since
(16)FK(x)=inf∑‍i=1nf(xi-k),wherek∈K,
then, for any constant α, there exists {kα} such that
(17)∑‍i=1nf(xi-kα)≤∑‍i=1nf(xi-k)+1α.
ButK is w-compact; then, there exists a subnet {kαβ} such that kαβ→k0. Thus,
(18)xi-kαβ⟶xi-k0,∀i=1,2,…,n.
Sincef is continuous, then
(19)∑‍i=1nf(xi-kαβ)≤∑‍i=1nf(xi-k)+1α.
Also,(20)∑‍i=1nf(xi-k0)=liminf∑‍i=1nf(xi-kαβ)≤∑‍i=1nf(xi-k).
Hence,k0∈PKf(x).For a subsetK of X, let us define KF^ to be such that
(21)KF^={x=(x1,x2,…,xn)∈Xn:FK(x)=∑i=1nf(xi)}.Example 12.
ConsiderX=(ℝ2)2 and K={((x1,y1),(x2,y2)):xi=yi,foralli=1,2}. Let f(x,y)=x2+y2; then, one can see that
(22)KF^={((x1,-x1),(x2,-x2))}.Using the previous definition ofKF^, we prove the following theorem characterizing f-simultaneously proximal subspaces.Theorem 13.
LetK be a vector subspace of X. Then, K is f-simultaneously proximal in X if and only if Xn=Dk+KF^, where DK={(k,k,…,k):k∈K}.Proof.
Suppose thatXn=Dk+KF^. Then, for x=(x1,x2,…,xn)∈Xn, there exists k1=(k0,k0,…,k0)∈DK and y=(y1,y2,…,yn)∈KF^ such that x=y+k1. Hence, x-k1=y∈KF^, and
(23)FK(y)=FK(x-k1)=∑‍i=1nf(xi-k0),
and so
(24)∑‍i=1nf(xi-k0)=infk∈K∑‍i=1nf(xi-k0-k)=infk′∈K∑‍i=1nf(xi-k′)=FK(x).
So, K is f-simultaneously proximal.
Conversely, suppose thatK is f-simultaneously proximal and x=(x1,x2,…,xn)∈Xn. Then, there exists k0∈K such that
(25)∑‍i=1nf(xi-k0)=infk∈K∑‍i=1nf(xi-k)=infk∈K∑‍i=1nf(xi-(k′+k0)),
where k=k′+k0. If k1=(k0,k0,…,k0)∈DK, then
(26)∑‍i=1nf(xi-k0)=FK(x-k1),
which implies that x-k1=k2∈KF^and Xn=Dk+KF^.Proposition 14.
LetX be a topological vector space and Kf-simultaneous proximal subset of X. Then,(1)
k0∈PKf(x) if and only if x-k0∈KF^;(2)
iff is symmetric (i.e., f(-x)=f(x) for all x∈X), then x∈KF^ if and only if -x∈KF^;(3)
ifx⊥FK, then x∈KF^, where x=(x1,x2,…,xn);(4)
ifx∈KF^ and αK=K, then x⊥FK, where x=(x1,x2,…,xn).Proof.
(
1
) Let k0∈PKf(x) if and only if ∑i=1nf(xi-k0)=inf{∑i=1nf(xi-k):k∈K}.
Thus,(27)∑‍i=1nf(xi-k0)=inf{∑‍i=1nf(xi-k0+k0-k):k∈K}=inf{∑‍i=1nf(xi-k0-k′):k′∈K},
which implies that x-k0∈KF^.
(
2
) Let x=(x1,x2,…,xn)∈KF^. Since f is symmetric, then
(28)∑‍i=1nf(-xi)=∑‍i=1nf(xi)=inf{∑‍i=1nf(xi-k):k∈K}=inf{∑‍i=1nf(-(-xi+k)):-k∈K}=inf{∑‍i=1nf(-xi+k):-k∈K}.
Hence,∑i=1nf(-xi)=inf{∑i=1nf(-xi+k):-k∈K}, which implies that
(29)-x=(-x1,-x2,…,-xn)∈KF^.
(
3
) Let x=(x1,x2,…,xn). Since x⊥FK, then
(30)∑‍i=1nf(xi)≤∑i=1nf(xi+αk)∀α∈ℝ,k∈K.=∑i=1nf(xi-(-αk))∀α∈ℝ,k∈K.=∑i=1nf(xi-k′),k′∈K.
So,(31)∑‍i=1nf(xi)≤∑‍i=1nf(xi-k′),k′∈K.
Hence,x=(x1,x2,…,xn)∈KF^.
(
4
) Let x∈KF^ and αK=K. Then,
(32)∑‍i=1nf(xi)=infk∈K∑‍i=1nf(xi-k)=infαk∈K∑‍i=1nf(xi-αk),sinceαK=K,=inf∑‍i=1nf(xi+(-αk)),∀k∈K.
Thus,(33)∑‍i=1nf(xi)≤∑‍i=1nf(xi+α′k),∀α′∈ℝ,∀k∈K.
Hence,x⊥FK.Theorem 15.
LetK be a vector subspace of X. If π(KF^)=Xn/DK, then K is f-simultaneously proximal, where π is the canonical map x→x+Dk.Proof.
Letπ(KF^)=Xn/DK and x=(x1,x2,…,xn)∈Xn. Then, x+DK=y+DK for some y∈KF^. Hence, x-y=k0 for some k0∈DK. Thus, x=y+k0∈KF^+Dk. Therefore, KF^+Dk=Xn. By Theorem 15, K is f-simultaneously proximal.


## 3.f-Simultaneous Approximation in Quotient Space

Definition 16.
LetK and  M be two vector subspaces of X such that M is closed and M⊂K. Suppose that f is a positive real-valued function defined on X. Then, a function f~:(X/M)n→ℝ can be defined as follows:
(34)f~(x1+M,x2+M,…,xn+M)=inf{∑‍i=1nf(xi+y):y∈M}
for each (x1,x2,…,xn)∈Xn.Theorem 17.
LetK and  M be two vector subspaces of X such that M⊂K. If k0 is a point of f-best simultaneous approximation to (x1,x2,…,xn) in K, then k0+M is an f~-best simultaneous approximation to (x1,x2,…,xn)+M inK/M.Proof.
Suppose thatk0+M is not f~-best simultaneous approximation to (x1+M,x2+M,…,xn+M) in K/M. Then,
(35)f~((xi-k0+M)i=1n)≰f~((xi-k+M)i=1n)
for at least k∈K, say k1∈K, such that
(36)f~((xi-k1+M)i=1n)<f~((xi-k0+M)i=1n).
Since
(37)f~((xi-k0+M)i=1n)=inf{∑‍i=1nf(xi-k0+y):y∈M}≤∑‍i=1nf(xi-k0),
we have
(38)f~((xi-k1+M)i=1n)<∑‍i=1nf(xi-k0).
Thus, for some m0∈M, we have
(39)∑‍i=1nf(xi-k1+m0)<∑‍i=1nf(xi-k0),
so,
(40)∑‍i=1nf(xi-(k1-m0))<∑‍i=1nf(xi-k0).
Since M⊂K implies that k1-m0∈K, therefore, k0 is not f-best simultaneous approximation to (x1,x2,…,xn) in K, which is a contradiction.Corollary 18.
LetK and M be two vector subspaces of X such that M⊂K. Then, if K is f-simultaneously proximal in X, then K/M is f~-simultaneously proximal in X/M.Proof.
IfK is f-simultaneously proximal in X, then there exists at least k0∈K such that k0 is f-best simultaneous approximation to (x1,x2,…,xn) in K. Thus by Theorem 11, k0+M is an f~-best simultaneous approximation to (x1,x2,…,xn)+M in K/M, so, K/M is f~-simultaneously proximal in X/M.Theorem 19.
LetK and M be two vector subspaces of X such that M⊂K. If M is f-simultaneously proximal in X and K/M is f~-simultaneously proximal in X/M, then K is f-simultaneously proximal in X.Proof.
SinceK/M is f~-simultaneously proximal in X/M, then there exists k0∈K such that k0+M is f~-best simultaneous approximation to (x1,x2,…,xn)+M from K/M, so,
(41)f~((xi-k0+M)i=1n)≤f~((xi-k+M)i=1n),∀k∈K,⇓f~((xi-k0+M)i=1n)=infm∈M∑‍i=1nf(xi-k0+m)≤infm∈M∑‍i=1nf(xi-k+m),
for all k∈K. Note that
(42)infm∈M∑‍i=1nf(xi-k0+m)=FM(x1-k0,x2-k0,…,xn-k0)≤FM(x1-k,x2-k,…,xn-k).
Since M is f-simultaneously proximal in X, then there exists m0∈M such that
(43)FM(x1-k0,x2-k0,…,xn-k0)=∑‍i=1nf(xi-k0-m0)≤∑‍i=1nf(xi-k-m),
for all m∈M and k∈K. So,
(44)∑‍i=1nf(xi-(k0+m0))≤∑‍i=1nf(xi-(k+m)),
for all m∈M and k∈K. Hence,
(45)∑‍i=1nf(xi-(k0+m0))=inf{∑‍i=1nf(xi-(k+m)):m∈M,k∈K}.
So, k0+m0 is an f-best simultaneous approximation to (x1,x2,…,xn) from K and K is f-simultaneously proximal in X.Theorem 20.
LetK and M be two vector subspaces of X such that M⊂K. If M is f-simultaneously proximal in X and K is f-simultaneously Chebyshev in X, then K/M is f~-simultaneously Chebyshev in X/M.Proof.
Suppose not, then there exists(x1,x2,…,xn)+M∈X/M, and k1+M, k2+M∈PK/Mf~((x1,x2,…,xn)+M) such that k1+M≠k2+M. Thus, k1-k2∉M. Since M is f-simultaneously proximal in X, then
(46)PMf(x1-k1,x2-k1,…,xn-k1)≠ϕ,PMf(x1-k2,x2-k2,…,xn-k2)≠ϕ.
Let m1∈PMf(x1-k1,x2-k1,…,xn-k1) and m2∈PMf(x1-k2,x2-k2,…,xn-k2). By Theorem 13, k1+m1 and k2+m2 are f-best simultaneous approximation to (x1,x2,…,xn) from K. Since K is f-simultaneously Chebyshev in X, then k1+m1=k2+m2, and, hence, k1-k2=m1-m2∈M, which is a contradiction.Theorem 21.
LetK and M be two vector subspaces of a topological vector space X. If M is f-simultaneously Chebyshev in X, then the following assertions are equivalent:(i)
K/M is f~-simultaneously Chebyshev in X/M;(ii)
K+M is simultaneously Chebyshev in X.Proof.
(i⇒ii) By hypothesis, (K+M)/M=K/M is f~-simultaneous Chebyshev. Assume that K+M is not f-simultaneous Chebyshev in X. Then, there exists x=(x1,…,xn)∈Xn which has two distinct f-best simultaneous approximations, say ℓ0 and ℓ1∈K+M. Thus, we have ℓ0 and ℓ1∈PK+Mf(x). Since M⊆K+M, we have that ℓ0+M and ℓ1+M∈P(K+M)/Mf(x+M)=PK/Mf(x+M). By hypothesis, K/M is f~-simultaneous Chebyshev, and so ℓ0+M=ℓ1+M. Then, there exists m0∈M∖{0} such that ℓ1=ℓ0+m0. Thus, we conclude that
(47)∑‍ni=1f((xi-ℓ0)-m0)=∑‍ni=1f(xi-ℓ1)=infm∈M{∑‍ni=1f(xi-(ℓ0+m))}≤{∑‍ni=1f((xi-ℓ0)-m)},∀m∈M=FM(x-ℓ0).
So,m0 and 0 are f-best simultaneous approximations to x-ℓ0 from M. Hence, M is not f-simultaneously Chebyshev. This is a contradiction.
(ii⇒i) Assume that (i) does not hold. Then, there exists x+M∈K/M which has two distinct f~-best simultaneous approximations, say k+M and k′+M∈K/M; thus, k-k′∉M. Since M is f-simultaneously proximal, so there exist f-best simultaneous approximations m and m′ to x-k and x-k′ from M, respectively. Therefore, we have m∈PMf(x-k) and m′∈PMf(x-k′). Since M⊆K+M, k+M and k′+M∈PK/Mf(x+M)=P(K+M)/Mf(x+M), so k+m and k′+m′∈PK+Mf(x). But K+M is f-simultaneously Chebyshev. Thus we get k+m=k′+m′, and therefore k-k′∈M. This is a contradiction.Definition 22.
A subsetK of X is called f-quasisimultaneously Chebyshev if PKf(x) is non-empty and f-compact set in X, for all x=(x1,x2,…,xn)∈Xn.Theorem 23.
Letf be a positive function, M an f-simultaneously proximal vector subspace of X, and Kf-quasisimultaneously Chebyshev of X such that M⊂K. Then, K/M is f~-quasi-simultaneously Chebyshev in Xn/M.Proof.
SinceK is f-simultaneously proximal in X, then by Corollary  12, K/M is f~-simultaneously proximal in X/M. Let x=(x1,x2,…,xn)∈Xn and (kn+M) a sequence in PK/Mf~(x+M). For every n, there exists mn∈M such that kn+mn=kn′∈PKf(x). But since M is a vector subspace, we have
(48)kn′+M=kn+mn+M=kn+M.
Since K is f-quasi-simultaneously Chebyshev of X, the sequence {kn} has a subsequence {kni} which is f-convergent to k0∈PKf(x), meaning that
(49)f(kni-k0)⟶0.
But
(50)f~(kni-k0+M)≤f(kni-k0)⟶0.
Hence,
(51)f~(kni-k0+M)⟶0.
Consequently,PK/Mf^(x+M) is f~-compact and K/M is f~-quasi-simultaneously Chebyshev. This completes the proof.



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