New Generalization of 𝑓-Best Simultaneous Approximation in Topological Vector Spaces

Abstract

Let 𝐾 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,…,𝑥_𝑛)∈𝑋^𝑛, there exists 𝑘₀∈𝐾 such that 𝐹_𝐾(𝑥)=∑^𝑛_𝑖=1𝑓(𝑥_𝑖−𝑘₀)=inf{∑^𝑛_𝑖=1𝑓(𝑥_𝑖−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.

1. Introduction

Let 𝐾 be a closed subset of a Hausdorff topological vector space 𝑋 and 𝑓 a real-valued continuous function on 𝑋. For 𝑥∈𝑋, set 𝐹_𝐾(𝑥)=inf_𝑘∈𝐾𝑓(𝑥−𝑘). A point 𝑘₀∈𝐾 is called 𝑓-best approximation to 𝑥 in 𝐾 if 𝐹_𝐾(𝑥)=𝑓(𝑥−𝑘₀). The set 𝑃^𝑓_𝐾(𝑥)={𝑘∈𝐾∶𝐹_𝐾(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, 𝑃^𝑓_𝐾(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. Let 𝐾 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,…,𝑥_𝑛)∈𝑋^𝑛 such that 𝐹_𝐾(𝑥)=inf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘∶𝑘∈𝐾=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀. (1)

The set of all 𝑓-best simultaneous approximations to 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 in 𝐾 is denoted by 𝑃^𝑓_𝐾(𝑥)=𝑘∈𝐾∶𝐹_𝐾(𝑥)=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘. (2)

The set 𝐾 is called 𝑓-simultaneously proximal (𝑓-simultaneously Chebyshev) if for each 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, 𝑃^𝑓_𝐾(𝑥)≠𝜙 (singleton). If 𝑛=1, simultaneous 𝑓-proximal is precisely 𝑓-proximal.

We remark that if 𝑓(𝑥)=‖𝑥‖, then the concept of 𝑓-best approximation is precisely the best approximation.

A set 𝐾 is said to be inf-compact at a point 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 [5] if each minimizing sequence in 𝐾 (i.e., ∑^𝑛_𝑖=1𝑓(𝑥_𝑖−𝑘_𝑛)→𝐹_𝐾(𝑥)) has a convergent subsequence in 𝐾. The set 𝐾 is called inf-compact if it is inf-compact at each 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛.

It is easy to see that if 𝐾 is compact or inf-compact, then 𝐾 is 𝑓-simultaneously proximal.

In this paper, we introduce the concept of 𝑓-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 of 𝑓-proximal sets in 𝑋. We begin with the following definitions.

Definition 2. A function 𝑓∶𝑋→ℝ is called absolutely homogeneous if 𝑓(𝛼𝑥)=|𝛼|𝑓(𝑥), for all 𝑥∈𝑋 and all 𝛼∈ℝ.

Definition 3. A subset 𝐾 of 𝑋 is called 𝑓-closed if for all sequences {𝑘_𝑚} of 𝐾 and for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, such that ∑^𝑛_𝑖=1𝑓(𝑥_𝑖−𝑘_𝑚)→0, we have 𝑥∈𝐾^𝑛.

Definition 4. A subset 𝐾 of 𝑋 is called 𝑓-compact if for every sequence {𝑘_𝑛} in 𝐾 there exist a subsequence {𝑘_𝑛𝑘} of {𝑘_𝑛} and 𝑘₀∈𝐾 such that 𝑓(𝑘_𝑛𝑘−𝑘₀)→0.

Definition 5. For 𝑥,𝑦∈𝑋, where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 and 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑋^𝑛, 𝑥 is said to be 𝑓-orthogonal to 𝑦 denoted by 𝑥⟂_𝑓𝑦, if ∑^𝑛_𝑖=1𝑓(𝑥_𝑖)≤∑^𝑛_𝑖=1𝑓(𝑥_𝑖+𝛼𝑦_𝑖) 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 that 𝐾 is 𝑤-compact if every net {𝑘_𝛼} in 𝐾 has a convergent subnet.

Theorem 7. Let 𝐾 be a subset of 𝑋. Then, one has the following.(1)𝐹_𝐾+𝑦(𝑥+𝑌)=𝐹_𝐾(𝑥), for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛), where 𝑌=(𝑦,𝑦,…,𝑦)∈𝑋^𝑛.(2)𝑃^𝑓_𝐾+𝑦(𝑥+𝑌)=𝑃^𝑓_𝐾(𝑥)+𝑦, for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛).(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, if 𝑓 is absolutely homogeneous function, then one has the following.(4)𝐹_𝛼𝐾(𝛼𝑥)=|𝛼|𝐹_𝐾(𝑥), for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 and 𝛼∈ℝ.(5)𝑃^𝑓_𝛼𝐾(𝛼𝑥)=𝛼𝑃^𝐹_𝐾(𝑥), for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 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) If 𝑓 is convex function and 𝐾 is a convex set, then 𝑃^𝑓_𝐾(𝑥) is convex.

Proof. (1) Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) and 𝑌=(𝑦,𝑦,…,𝑦)∈𝑋^𝑛. Then 𝐹_𝐾+𝑦(𝑥+𝑌)=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖+𝑦−𝜅+𝑦=𝐹_𝐾(𝑥). (3)
(2) The equation ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖+𝑦−𝑘+𝑦=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘 (4) implies that 𝑘₀+𝑦∈𝑃^𝑓_𝐾+𝑦(𝑥+𝑌) if and only if 𝑘₀∈𝑃^𝑓_𝐾(𝑥). Thus, 𝑃^𝑓_𝐾+𝑦(𝑥+𝑌)=𝑃^𝑓_𝐾(𝑥)+𝑦. (5)
(3) The proof follows immediately from part (2) above.
(4) Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, 𝛼∈ℝ. Then, 𝐹_𝛼𝐾(𝛼𝑥)=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝛼𝑥_𝑖=|𝛼|−𝛼𝑘inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘=|𝛼|𝐹_𝐾(𝑥). (6)
(5) If 𝛼=0, then we are done. If 𝛼≠0 and 𝑘₀∈𝑃^𝑓_𝛼𝐾(𝛼𝑥), then 𝑘₀∈𝛼𝐾 and ^𝑛_𝑖=1𝑓𝛼𝑥_𝑖−𝑘₀=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝛼𝑥_𝑖−𝛼𝑘. (7) This implies that ^𝑛_𝑖=1𝑓𝑥_𝑖−1𝛼𝑘₀=𝐹_𝐾(𝑥), (8) which implies that (1/𝛼)𝑘₀∈𝑃^𝑓_𝐾(𝑥).
(6) The proof follows immediately from part (5) above.
(7) Let 𝑘₁, 𝑘₂∈𝑃^𝑓_𝐾(𝑥). Since 𝐾 is convex, then 𝜆𝑘₁−(1−𝜆)𝑘₂∈𝐾. We must show that 𝜆𝑘₁−(1−𝜆)𝑘₂∈𝑃^𝑓_𝐾(𝑥); that is, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝜆𝑘₁−(1−𝜆)𝑘₂=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘. (9) So, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝜆𝑘₁−(1−𝜆)𝑘₂=^𝑛_𝑖=1𝑓𝜆𝑥_𝑖−𝑘₁+(1−𝜆)𝑥_𝑖−𝑘₂=𝜆^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₁+(1−𝜆)^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₂=𝜆𝐹_𝐾(𝑥)+(1−𝜆)𝐹_𝐾(𝑥)=𝐹_𝐾(𝑥)=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘, (10) which implies that 𝑃^𝑓_𝐾(𝑥) is convex.

Example 8. Let 𝑋=ℝ² and 𝐾={(𝑥₁,𝑥₂)∈ℝ²∶𝑥²₁+𝑥²₂≤4}, and let 𝑓(𝑥,𝑦)=𝑥²−𝑦². If 𝑧=((0,0),(0,1))∈𝑋², then one can show that 𝐹_𝐾(𝑧)=𝑓(0,1/2)=−1/4.

Theorem 9. Let 𝑓 be an absolutely homogeneous real-valued function on 𝑋 and 𝑀 a vector subspace of 𝑋. Then,(1)𝐹_𝑀(𝛼𝑥)=|𝛼|𝐹_𝑀(𝑥), for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, 𝛼∈ℝ−{0};(2)𝑃^𝑓_𝑀(𝛼𝑥)=𝛼𝑃^𝑓_𝑀(𝑥), for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, 𝛼∈ℝ−{0}.

Proof. (1) Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛). Then, 𝐹_𝑀(𝛼𝑥)=inf^𝑛_{𝑚∈𝑀𝑖=1}𝑓𝛼𝑥_𝑖=|𝛼|−𝑚inf_𝑚′^𝑛_{∈𝑀𝑖=1}𝑓𝑥_𝑖−𝑚^′=|𝛼|𝐹_𝑀(𝑥). (11)
(2) Let 𝑚₀∈𝑃^𝑓_𝑀(𝛼𝑥). Then, ^𝑛_𝑖=1𝑓𝛼𝑥_𝑖−𝑚₀=inf^𝑛_{𝑚∈𝑀𝑖=1}𝑓𝛼𝑥_𝑖−𝑚 (12) if and only if ^𝑛_𝑖=1𝑓𝑥_𝑖−1𝛼𝑚₀=inf_𝑚′^𝑛_∈𝑀𝑓𝑥_𝑖−𝑚^′=𝐹_𝑀(𝑥), (13) for all 𝛼∈ℝ−{0}, which implies that (1/𝛼)𝑚₀∈𝑃^𝑓_𝑀(𝑥), so, 𝑚₀∈𝛼𝑃^𝑓_𝑀(𝑥).

Theorem 10. Let 𝑓 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 . Since 𝑓 is a positive function, then ∑^𝑛_𝑖=1𝑓(𝑥_𝑖)≥0 for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛. Let {𝑘_𝑚} be a sequence of 𝐾 and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, such that ∑^𝑛_𝑖=1𝑓(𝑥_𝑖−𝑘_𝑚)→0. This implies that 𝐹_𝐾(𝑥)=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘≤^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘_𝑚⟶0. (14) Since 𝐾 is 𝑓-simultaneously proximal, then there exists 𝑘₀∈𝐾 such that 𝐹_𝐾(𝑥)=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=0. (15) Hence, for all𝑖=1,2,…,𝑛, 𝑓(𝑥_𝑖−𝑘₀)=0.Using the assumption it follows that 𝑥_𝑖−𝑘₀=0, and, hence, 𝑥_𝑖=𝑘₀∈𝐾. Consequently, 𝑥∈𝐾^𝑛 and 𝐾 is 𝑓-closed.

Theorem 11. Let 𝑋 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. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛. Since 𝐹_𝐾(𝑥)=inf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘,where𝑘∈𝐾, (16) then, for any constant 𝛼, there exists {𝑘_𝛼} such that ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘_𝛼≤^𝑛_𝑖=1𝑓𝑥_𝑖+1−𝑘𝛼. (17)
But 𝐾 is 𝑤-compact; then, there exists a subnet {𝑘_𝛼𝛽} such that 𝑘_𝛼𝛽→𝑘₀. Thus, 𝑥_𝑖−𝑘_𝛼𝛽⟶𝑥_𝑖−𝑘₀,∀𝑖=1,2,…,𝑛. (18)
Since 𝑓 is continuous, then ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘_𝛼𝛽≤^𝑛_𝑖=1𝑓𝑥_𝑖+1−𝑘𝛼. (19)
Also, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=liminf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘_𝛼𝛽≤^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘. (20)
Hence, 𝑘₀∈𝑃^𝑓_𝐾(𝑥).

For a subset 𝐾 of 𝑋, let us define 𝐾_𝐹 to be such that 𝐾_𝐹=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋^𝑛∶𝐹_𝐾(𝑥)=^𝑛_𝑖=1𝑓𝑥_𝑖. (21)

Example 12. Consider 𝑋=(ℝ²)² and 𝐾={((𝑥₁,𝑦₁),(𝑥₂,𝑦₂))∶𝑥_𝑖=𝑦_𝑖,forall𝑖=1,2}. Let 𝑓(𝑥,𝑦)=𝑥²+𝑦²; then, one can see that 𝐾_𝐹=𝑥₁,−𝑥₁,𝑥₂,−𝑥₂. (22)

Using the previous definition of 𝐾_𝐹, we prove the following theorem characterizing 𝑓-simultaneously proximal subspaces.

Theorem 13. Let 𝐾 be a vector subspace of 𝑋. Then, 𝐾 is 𝑓-simultaneously proximal in 𝑋 if and only if 𝑋^𝑛=𝐷_𝑘+𝐾_𝐹, where 𝐷_𝐾={(𝑘,𝑘,…,𝑘)∶𝑘∈𝐾}.

Proof. Suppose that 𝑋^𝑛=𝐷_𝑘+𝐾_𝐹. Then, for 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛, there exists 𝑘₁=(𝑘₀,𝑘₀,…,𝑘₀)∈𝐷_𝐾 and 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝐾_𝐹 such that 𝑥=𝑦+𝑘₁. Hence, 𝑥−𝑘₁=𝑦∈𝐾_𝐹, and 𝐹_𝐾𝑦=𝐹_𝐾𝑥−𝑘₁=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀, (23) and so ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘₀−𝑘=inf_𝑘′^𝑛_{∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘^′=𝐹_𝐾(𝑥). (24) So, 𝐾 is 𝑓-simultaneously proximal.
Conversely, suppose that 𝐾 is 𝑓-simultaneously proximal and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛. Then, there exists 𝑘₀∈𝐾 such that ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘^′+𝑘₀, (25) where 𝑘=𝑘^′+𝑘₀. If 𝑘₁=(𝑘₀,𝑘₀,…,𝑘₀)∈𝐷_𝐾, then ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=𝐹_𝐾𝑥−𝑘₁, (26) which implies that 𝑥−𝑘₁=𝑘₂∈𝐾_𝐹and 𝑋^𝑛=𝐷_𝑘+𝐾_𝐹.

Proposition 14. Let 𝑋 be a topological vector space and 𝐾𝑓-simultaneous proximal subset of 𝑋. Then,(1)𝑘₀∈𝑃^𝑓_𝐾(𝑥) if and only if 𝑥−𝑘₀∈𝐾_𝐹;(2)if 𝑓 is symmetric (i.e., 𝑓(−𝑥)=𝑓(𝑥) for all 𝑥∈𝑋), then 𝑥∈𝐾_𝐹 if and only if −𝑥∈𝐾_𝐹;(3)if 𝑥⟂_𝐹𝐾, then 𝑥∈𝐾_𝐹, where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛);(4)if 𝑥∈𝐾_𝐹 and 𝛼𝐾=𝐾, then 𝑥⟂_𝐹𝐾, where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛).

Proof. (1) Let 𝑘₀∈𝑃^𝑓_𝐾(𝑥) if and only if ∑^𝑛_𝑖=1𝑓(𝑥_𝑖−𝑘₀)=inf{∑^𝑛_𝑖=1𝑓(𝑥_𝑖−𝑘)∶𝑘∈𝐾}.
Thus, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀=inf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀+𝑘₀−𝑘∶𝑘∈𝐾=inf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀−𝑘^′∶𝑘^′∈𝐾, (27) which implies that 𝑥−𝑘₀∈𝐾_𝐹.
(2) Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝐾_𝐹. Since 𝑓 is symmetric, then ^𝑛_𝑖=1𝑓−𝑥_𝑖=^𝑛_𝑖=1𝑓𝑥_𝑖=inf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘∶𝑘∈𝐾=inf^𝑛_𝑖=1𝑓−−𝑥_𝑖+𝑘∶−𝑘∈𝐾=inf^𝑛_𝑖=1𝑓−𝑥_𝑖+𝑘∶−𝑘∈𝐾. (28)
Hence, ∑^𝑛_𝑖=1𝑓(−𝑥_𝑖)=inf{∑^𝑛_𝑖=1𝑓(−𝑥_𝑖+𝑘)∶−𝑘∈𝐾}, which implies that −𝑥=−𝑥₁,−𝑥₂,…,−𝑥_𝑛∈𝐾_𝐹. (29)
(3) Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛). Since 𝑥⟂_𝐹𝐾, then ^𝑛_𝑖=1𝑓𝑥_𝑖≤^𝑛_𝑖=1𝑓𝑥_𝑖=+𝛼𝑘∀𝛼∈ℝ,𝑘∈𝐾._𝑛𝑖=1𝑓𝑥_𝑖−()=−𝛼𝑘∀𝛼∈ℝ,𝑘∈𝐾._𝑛𝑖=1𝑓𝑥_𝑖−𝑘^′,𝑘^′∈𝐾. (30)
So, ^𝑛_𝑖=1𝑓𝑥_𝑖≤^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘^′,𝑘^′∈𝐾. (31)
Hence, 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝐾_𝐹.
(4) Let 𝑥∈𝐾_𝐹 and 𝛼𝐾=𝐾. Then, ^𝑛_𝑖=1𝑓𝑥_𝑖=inf^𝑛_{𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝑘=inf^𝑛_{𝛼𝑘∈𝐾𝑖=1}𝑓𝑥_𝑖−𝛼𝑘,since𝛼𝐾=𝐾,=inf^𝑛_𝑖=1𝑓𝑥_𝑖+(−𝛼𝑘),∀𝑘∈𝐾. (32)
Thus, ^𝑛_𝑖=1𝑓𝑥_𝑖≤^𝑛_𝑖=1𝑓𝑥_𝑖+𝛼^′𝑘,∀𝛼^′∈ℝ,∀𝑘∈𝐾. (33)
Hence, 𝑥⟂_𝐹𝐾.

Theorem 15. Let 𝐾 be a vector subspace of 𝑋. If 𝜋(𝐾_𝐹)=𝑋^𝑛/𝐷_𝐾, then 𝐾 is 𝑓-simultaneously proximal, where 𝜋 is the canonical map 𝑥→𝑥+𝐷_𝑘.

Proof. Let 𝜋(𝐾_𝐹)=𝑋^𝑛/𝐷_𝐾 and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛. Then, 𝑥+𝐷_𝐾=𝑦+𝐷_𝐾 for some 𝑦∈𝐾_𝐹. Hence, 𝑥−𝑦=𝑘₀ for some 𝑘₀∈𝐷_𝐾. Thus, 𝑥=𝑦+𝑘₀∈𝐾_𝐹+𝐷_𝑘. Therefore, 𝐾_𝐹+𝐷_𝑘=𝑋^𝑛. By Theorem 15, 𝐾 is 𝑓-simultaneously proximal.

3. 𝑓-Simultaneous Approximation in Quotient Space

Definition 16. Let 𝐾 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: 𝑓𝑥₁+𝑀,𝑥₂+𝑀,…,𝑥_𝑛+𝑀=inf^𝑛_𝑖=1𝑓𝑥_𝑖+𝑦∶𝑦∈𝑀 (34) for each (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛.

Theorem 17. Let 𝐾 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,…,𝑥_𝑛) in 𝐾, then 𝑘₀+𝑀 is an 𝑓-best simultaneous approximation to (𝑥₁,𝑥₂,…,𝑥_𝑛)+𝑀 in𝐾/𝑀.

Proof. Suppose that 𝑘₀+𝑀 is not 𝑓-best simultaneous approximation to (𝑥₁+𝑀,𝑥₂+𝑀,…,𝑥_𝑛+𝑀) in 𝐾/𝑀. Then, 𝑓𝑥_𝑖−𝑘₀+𝑀^𝑛_𝑖=1≰𝑓𝑥_𝑖−𝑘+𝑀^𝑛_𝑖=1 (35) for at least 𝑘∈𝐾, say 𝑘₁∈𝐾, such that 𝑓𝑥_𝑖−𝑘₁+𝑀^𝑛_𝑖=1<𝑓𝑥_𝑖−𝑘₀+𝑀^𝑛_𝑖=1. (36) Since 𝑓𝑥_𝑖−𝑘₀+𝑀^𝑛_𝑖=1=inf^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀≤+𝑦∶𝑦∈𝑀_𝑛𝑖=1𝑓𝑥_𝑖−𝑘₀, (37) we have 𝑓𝑥_𝑖−𝑘₁+𝑀^𝑛_𝑖=1<^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀. (38) Thus, for some 𝑚₀∈𝑀, we have ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₁+𝑚₀<^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀, (39) so, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₁−𝑚₀<^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀. (40) Since 𝑀⊂𝐾 implies that 𝑘₁−𝑚₀∈𝐾, therefore, 𝑘₀ is not 𝑓-best simultaneous approximation to (𝑥₁,𝑥₂,…,𝑥_𝑛) in 𝐾, which is a contradiction.

Corollary 18. Let 𝐾 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. If 𝐾 is 𝑓-simultaneously proximal in 𝑋, then there exists at least 𝑘₀∈𝐾 such that 𝑘₀ is 𝑓-best simultaneous approximation to (𝑥₁,𝑥₂,…,𝑥_𝑛) in 𝐾. Thus by Theorem 11, 𝑘₀+𝑀 is an 𝑓-best simultaneous approximation to (𝑥₁,𝑥₂,…,𝑥_𝑛)+𝑀 in 𝐾/𝑀, so, 𝐾/𝑀 is 𝑓-simultaneously proximal in 𝑋/𝑀.

Theorem 19. Let 𝐾 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. Since 𝐾/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,…,𝑥_𝑛)+𝑀 from 𝐾/𝑀, so, 𝑓𝑥_𝑖−𝑘₀+𝑀^𝑛_𝑖=1≤𝑓𝑥_𝑖−𝑘+𝑀^𝑛_𝑖=1⇓𝑓𝑥,∀𝑘∈𝐾,_𝑖−𝑘₀+𝑀^𝑛_𝑖=1=inf^𝑛_{𝑚∈𝑀𝑖=1}𝑓𝑥_𝑖−𝑘₀+𝑚≤inf^𝑛_{𝑚∈𝑀𝑖=1}𝑓𝑥_𝑖−𝑘+𝑚, (41) for all 𝑘∈𝐾. Note that inf^𝑛_{𝑚∈𝑀𝑖=1}𝑓𝑥_𝑖−𝑘₀+𝑚=𝐹_𝑀𝑥₁−𝑘₀,𝑥₂−𝑘₀,…,𝑥_𝑛−𝑘₀≤𝐹_𝑀𝑥₁−𝑘,𝑥₂−𝑘,…,𝑥_𝑛−𝑘. (42) Since 𝑀 is 𝑓-simultaneously proximal in 𝑋, then there exists 𝑚₀∈𝑀 such that 𝐹_𝑀𝑥₁−𝑘₀,𝑥₂−𝑘₀,…,𝑥_𝑛−𝑘₀=^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀−𝑚₀≤^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘−𝑚, (43) for all 𝑚∈𝑀 and 𝑘∈𝐾. So, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀+𝑚₀≤^𝑛_𝑖=1𝑓𝑥_𝑖−(𝑘+𝑚), (44) for all 𝑚∈𝑀 and 𝑘∈𝐾. Hence, ^𝑛_𝑖=1𝑓𝑥_𝑖−𝑘₀+𝑚₀=inf^𝑛_𝑖=1𝑓𝑥_𝑖−(𝑘+𝑚)∶𝑚∈𝑀,𝑘∈𝐾. (45) So, 𝑘₀+𝑚₀ is an 𝑓-best simultaneous approximation to (𝑥₁,𝑥₂,…,𝑥_𝑛) from 𝐾 and 𝐾 is 𝑓-simultaneously proximal in 𝑋.

Theorem 20. Let 𝐾 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,…,𝑥_𝑛)+𝑀∈𝑋/𝑀, and 𝑘₁+𝑀, 𝑘₂𝑓+𝑀∈𝑃_𝐾/𝑀((𝑥₁,𝑥₂,…,𝑥_𝑛)+𝑀) such that 𝑘₁+𝑀≠𝑘₂+𝑀. Thus, 𝑘₁−𝑘₂∉𝑀. Since 𝑀 is 𝑓-simultaneously proximal in 𝑋, then 𝑃^𝑓_𝑀𝑥₁−𝑘₁,𝑥₂−𝑘₁,…,𝑥_𝑛−𝑘₁𝑃≠𝜙,_𝑓𝑀𝑥₁−𝑘₂,𝑥₂−𝑘₂,…,𝑥_𝑛−𝑘₂≠𝜙. (46) Let 𝑚₁∈𝑃^𝑓_𝑀(𝑥₁−𝑘₁,𝑥₂−𝑘₁,…,𝑥_𝑛−𝑘₁) and 𝑚₂∈𝑃^𝑓_𝑀(𝑥₁−𝑘₂,𝑥₂−𝑘₂,…,𝑥_𝑛−𝑘₂). By Theorem 13, 𝑘₁+𝑚₁ and 𝑘₂+𝑚₂ are 𝑓-best simultaneous approximation to (𝑥₁,𝑥₂,…,𝑥_𝑛) 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. Let 𝐾 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,…,𝑥_𝑛)∈𝑋^𝑛 which has two distinct 𝑓-best simultaneous approximations, say ℓ₀ and ℓ₁∈𝐾+𝑀. Thus, we have ℓ₀ and ℓ₁∈𝑃^𝑓_𝐾+𝑀(𝑥). Since 𝑀⊆𝐾+𝑀, we have that ℓ₀+𝑀 and ℓ₁+𝑀∈𝑃^𝑓_{(𝐾+𝑀)/𝑀}(𝑥+𝑀)=𝑃^𝑓_𝐾/𝑀(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 ^𝑛_𝑖=1𝑓𝑥_𝑖−ℓ₀−𝑚₀=^𝑛_𝑖=1𝑓𝑥_𝑖−ℓ₁=inf_𝑚∈𝑀^𝑛_𝑖=1𝑓𝑥_𝑖−ℓ₀≤+𝑚_𝑛𝑖=1𝑓𝑥_𝑖−ℓ₀−𝑚,∀𝑚∈𝑀=𝐹_𝑀𝑥−ℓ₀. (47)
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∈𝑃^𝑓_𝑀(𝑥−𝑘) and 𝑚^′∈𝑃^𝑓_𝑀(𝑥−𝑘^′). Since 𝑀⊆𝐾+𝑀, 𝑘+𝑀 and 𝑘^′+𝑀∈𝑃^𝑓_𝐾/𝑀(𝑥+𝑀)=𝑃^𝑓_{(𝐾+𝑀)/𝑀}(𝑥+𝑀), so 𝑘+𝑚 and 𝑘^′+𝑚^′∈𝑃^𝑓_𝐾+𝑀(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 subset 𝐾 of 𝑋 is called 𝑓-quasisimultaneously Chebyshev if 𝑃^𝑓_𝐾(𝑥) is non-empty and 𝑓-compact set in 𝑋, for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛.

Theorem 23. Let 𝑓 be a positive function, 𝑀 an 𝑓-simultaneously proximal vector subspace of 𝑋, and K 𝑓-quasisimultaneously Chebyshev of 𝑋 such that 𝑀⊂𝐾. Then, 𝐾/𝑀 is 𝑓-quasi-simultaneously Chebyshev in 𝑋^𝑛/𝑀.

Proof. Since 𝐾 is 𝑓-simultaneously proximal in 𝑋, then by Corollary  12, 𝐾/𝑀 is 𝑓-simultaneously proximal in 𝑋/𝑀. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋^𝑛 and (𝑘_𝑛+𝑀) a sequence in 𝑃𝑓_𝐾/𝑀(𝑥+𝑀). For every 𝑛, there exists 𝑚_𝑛∈𝑀 such that 𝑘_𝑛+𝑚_𝑛=𝑘^′_𝑛∈𝑃^𝑓_𝐾(𝑥). But since 𝑀 is a vector subspace, we have 𝑘^′_𝑛+𝑀=𝑘_𝑛+𝑚_𝑛+𝑀=𝑘_𝑛+𝑀. (48) Since 𝐾 is 𝑓-quasi-simultaneously Chebyshev of 𝑋, the sequence {𝑘_𝑛} has a subsequence {𝑘_𝑛𝑖} which is 𝑓-convergent to 𝑘₀∈𝑃^𝑓_𝐾(𝑥), meaning that 𝑓𝑘_𝑛𝑖−𝑘₀⟶0. (49) But 𝑓𝑘_𝑛𝑖−𝑘₀+𝑀≤𝑓𝑘_𝑛𝑖−𝑘₀⟶0. (50) Hence, 𝑓𝑘_𝑛𝑖−𝑘₀+𝑀⟶0. (51)
Consequently, 𝑃𝑓_𝐾/𝑀(𝑥+𝑀) is 𝑓-compact and 𝐾/𝑀 is 𝑓-quasi-simultaneously Chebyshev. This completes the proof.

Acknowledgments

The authors would like to express their appreciation and gratitude to the editor and the anonymous referees for their comments and suggestions for this paper. Also, the authors wish to thank Professor Sharifa Al-Sharif for her valuable suggestions and comments.