Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 978738, 7 pages
http://dx.doi.org/10.1155/2013/978738
Research Article
New Generalization of 𝑓-Best Simultaneous Approximation in
Topological Vector Spaces
Mahmoud Rawashdeh and Sarah Khalil
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan
Correspondence should be addressed to Mahmoud Rawashdeh; msalrawashdeh@just.edu.jo
Received 20 January 2013; Revised 23 April 2013; Accepted 23 April 2013
Academic Editor: Naseer Shahzad
Copyright © 2013 M. Rawashdeh and S. Khalil. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
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
𝑛
𝑛
each 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 , there exists 𝑘0 ∈ 𝐾 such that 𝐹𝐾 (𝑥) = ∑𝑖=1 𝑓(𝑥𝑖 − 𝑘0 ) = inf{∑𝑖=1 𝑓(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.
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 𝐹𝐾 (𝑥) = inf 𝑘∈𝐾 𝑓(𝑥 − 𝑘). A point 𝑘0 ∈ 𝐾 is
called 𝑓-best approximation to 𝑥 in 𝐾 if 𝐹𝐾 (𝑥) = 𝑓(𝑥 − 𝑘0 ).
𝑓
The set 𝑃𝐾 (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-pro𝑓
ximal (𝑓-Chebyshev) if for each 𝑥 ∈ 𝑋, 𝑃𝐾 (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 𝑓best simultaneous approximation in 𝐾 if there exists 𝑥 =
(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 such that
𝑛
𝑛
𝑖=1
𝑖=1
𝐹𝐾 (𝑥) = inf {∑ 𝑓 (𝑥𝑖 − 𝑘) : 𝑘 ∈ 𝐾} =∑ 𝑓 (𝑥𝑖 − 𝑘0 ) .
(1)
The set of all 𝑓-best simultaneous approximations to 𝑥 =
(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 in 𝐾 is denoted by
𝑓
𝑛
𝑃𝐾 (𝑥) = {𝑘 ∈ 𝐾 : 𝐹𝐾 (𝑥) =∑ 𝑓 (𝑥𝑖 − 𝑘)} .
(2)
𝑖=1
The set 𝐾 is called 𝑓-simultaneously proximal (𝑓-simultaneously Chebyshev) if for each 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 ,
𝑓
𝑃𝐾 (𝑥) ≠ 𝜙 (singleton). If 𝑛 = 1, simultaneous 𝑓-proximal is
precisely 𝑓-proximal.
2
Abstract and Applied Analysis
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 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 [5] if each minimizing sequence in 𝐾
(i.e., ∑𝑛𝑖=1 𝑓(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 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.
𝑓
(5) 𝑃𝛼𝐾 (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.
𝑓
(7) If 𝑓 is convex function and 𝐾 is a convex set, then 𝑃𝐾 (𝑥)
is convex.
Proof. (1) Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) and 𝑌 = (𝑦, 𝑦, . . . , 𝑦) ∈ 𝑋𝑛 .
Then
𝑛
𝐹𝐾+𝑦 (𝑥 + 𝑌) = inf ∑ 𝑓 ((𝑥𝑖 + 𝑦) − (𝜅 + 𝑦)) = 𝐹𝐾 (𝑥) .
𝑘∈𝐾
(3)
(2) The equation
𝑛
𝑛
∑𝑓 (𝑥𝑖 − 𝑘0 ) = inf ∑ 𝑓 ((𝑥𝑖 + 𝑦) − (𝑘 + 𝑦))
2. 𝑓-Simultaneous Approximation
𝑘∈𝐾
𝑖=1
Definition 3. A subset 𝐾 of 𝑋 is called 𝑓-closed if for all
sequences {𝑘𝑚 } of 𝐾 and for all 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 ,
such that ∑𝑛𝑖=1 𝑓(𝑥𝑖 − 𝑘𝑚 ) → 0, we have 𝑥 ∈ 𝐾𝑛 .
Definition 4. A subset 𝐾 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. For 𝑥, 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⊥𝑓 𝑦, if ∑𝑛𝑖=1 𝑓(𝑥𝑖 ) ≤ ∑𝑛𝑖=1 𝑓(𝑥𝑖 + 𝛼𝑦𝑖 ) for every
scalar 𝛼 ∈ R. Also, 𝑥 is said to be 𝑓-orthogonal to a set 𝐾 if
𝑥⊥𝑓 𝑘, for all 𝑘 ∈ 𝐾.
Definition 6. We say that 𝐾 is 𝑤-compact if every net {𝑘𝛼 } in
𝐾 has a convergent subnet.
(2)
𝑓
𝑃𝐾+𝑦 (𝑥
+ 𝑌) =
𝑓
𝑃𝐾 (𝑥)
𝑘∈𝐾
Moreover, if 𝑓 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.
𝑖=1
𝑓
𝑓
implies that 𝑘0 + 𝑦 ∈ 𝑃𝐾+𝑦 (𝑥 + 𝑌) if and only if 𝑘0 ∈ 𝑃𝐾 (𝑥).
Thus,
𝑓
𝑓
𝑃𝐾+𝑦 (𝑥 + 𝑌) = 𝑃𝐾 (𝑥) + 𝑦.
(5)
(3) The proof follows immediately from part (2) above.
(4) Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 , 𝛼 ∈ R. Then,
𝑛
𝐹𝛼𝐾 (𝛼𝑥) = inf ∑ 𝑓 (𝛼𝑥𝑖 − 𝛼𝑘)
𝑘∈𝐾
𝑖=1
(6)
𝑛
= |𝛼| inf ∑ 𝑓 (𝑥𝑖 − 𝑘) = |𝛼| 𝐹𝐾 (𝑥) .
𝑘∈𝐾
𝑖=1
𝑓
(5) If 𝛼 = 0, then we are done. If 𝛼 ≠ 0 and 𝑘0 ∈ 𝑃𝛼𝐾 (𝛼𝑥),
then 𝑘0 ∈ 𝛼𝐾 and
𝑛
𝑛
∑ 𝑓 (𝛼𝑥𝑖 − 𝑘0 ) = inf ∑ 𝑓 (𝛼𝑥𝑖 − 𝛼𝑘) .
𝑘∈𝐾
𝑖=1
(7)
𝑖=1
This implies that
𝑛
∑ 𝑓 (𝑥𝑖 −
+ 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.
(4)
= inf ∑ 𝑓 (𝑥𝑖 − 𝑘)
Theorem 7. Let 𝐾 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 .
𝑖=1
𝑛
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.
𝑖=1
𝑖=1
1
𝑘 ) = 𝐹𝐾 (𝑥) ,
𝛼 0
(8)
𝑓
which implies that (1/𝛼)𝑘0 ∈ 𝑃𝐾 (𝑥).
(6) The proof follows immediately from part (5) above.
𝑓
(7) Let 𝑘1 , 𝑘2 ∈ 𝑃𝐾 (𝑥). Since 𝐾 is convex, then 𝜆𝑘1 − (1 −
𝑓
𝜆)𝑘2 ∈ 𝐾. We must show that 𝜆𝑘1 − (1 − 𝜆)𝑘2 ∈ 𝑃𝐾 (𝑥); that is,
𝑛
𝑛
∑ 𝑓 (𝑥𝑖 − (𝜆𝑘1 − (1 − 𝜆) 𝑘2 )) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘) .
𝑖=1
𝑘∈𝐾
𝑖=1
(9)
Abstract and Applied Analysis
3
𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 , such that ∑𝑛𝑖=1 𝑓(𝑥𝑖 − 𝑘𝑚 ) → 0.
This implies that
So,
𝑛
∑ 𝑓 (𝑥𝑖 − (𝜆𝑘1 − (1 − 𝜆) 𝑘2 ))
𝑘∈𝐾
𝑛
=∑ 𝑓 (𝜆 (𝑥𝑖 − 𝑘1 ) + (1 − 𝜆) (𝑥𝑖 − 𝑘2 ))
𝑛
𝑛
𝑛
𝑖=1
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.
is convex.
Example 8. Let 𝑋 = 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. Let 𝑓 be an absolutely homogeneous real-valued
function on 𝑋 and 𝑀 a vector subspace of 𝑋. Then,
Proof. Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 . Since
𝑛
𝐹𝐾 (𝑥) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘) ,
Proof. (1) Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ). Then,
𝑖=1
𝑖=1
1
.
𝛼
(17)
But 𝐾 is 𝑤-compact; then, there exists a subnet {𝑘𝛼𝛽 } such
that 𝑘𝛼𝛽 → 𝑘0 . Thus,
∀𝑖 = 1, 2, . . . , 𝑛.
(18)
Since 𝑓 is continuous, then
𝑖=1
(11)
𝑚 ∈𝑀 𝑖=1
𝑛
𝑛
𝑖=1
𝑖=1
∑ 𝑓 (𝑥𝑖 − 𝑘𝛼𝛽 ) ≤∑ 𝑓 (𝑥𝑖 − 𝑘) +
= |𝛼| inf
∑ 𝑓 (𝑥𝑖 − 𝑚󸀠 ) = |𝛼| 𝐹𝑀 (𝑥) .
󸀠
1
.
𝛼
(19)
Also,
Then,
𝑛
𝑛
𝑚∈𝑀
𝑛
𝑛
𝑖=1
𝑖=1
∑ 𝑓 (𝑥𝑖 − 𝑘0 ) = lim inf ∑ 𝑓 (𝑥𝑖 − 𝑘𝛼𝛽 )
∑ 𝑓 (𝛼𝑥𝑖 − 𝑚0 ) = inf ∑ 𝑓 (𝛼𝑥𝑖 − 𝑚)
𝑖=1
𝑛
𝑥𝑖 − 𝑘𝛼𝛽 󳨀→ 𝑥𝑖 − 𝑘0 ,
𝑛
(2) Let 𝑚0 ∈
𝑛
∑ 𝑓 (𝑥𝑖 − 𝑘𝛼 ) ≤∑ 𝑓 (𝑥𝑖 − 𝑘) +
𝑛
𝐹𝑀 (𝛼𝑥) = inf ∑ 𝑓 (𝛼𝑥𝑖 − 𝑚)
(16)
then, for any constant 𝛼, there exists {𝑘𝛼 } such that
𝑓
(2) 𝑃𝑀(𝛼𝑥) = 𝛼𝑃𝑀(𝑥), for all 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 ,
𝛼 ∈ R − {0}.
where 𝑘 ∈ 𝐾,
𝑖=1
(1) 𝐹𝑀(𝛼𝑥) = |𝛼|𝐹𝑀(𝑥), for all 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 ,
𝛼 ∈ R − {0};
𝑓
𝑃𝑀(𝛼𝑥).
(15)
Hence, for all 𝑖 = 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.
= 𝐹𝐾 (𝑥) =∑ 𝑓 (𝑥𝑖 − 𝑘) ,
𝑚∈𝑀
(14)
𝑖=1
= 𝜆𝐹𝐾 (𝑥) + (1 − 𝜆) 𝐹𝐾 (𝑥)
𝑓
𝑖=1
𝐹𝐾 (𝑥) =∑ 𝑓 (𝑥𝑖 − 𝑘0 ) = 0.
𝑖=1
which implies that
𝑖=1
𝑛
(10)
= 𝜆 ∑ 𝑓 (𝑥𝑖 − 𝑘1 ) + (1 − 𝜆) ∑ 𝑓 (𝑥𝑖 − 𝑘2 )
𝑓
𝑃𝐾 (𝑥)
𝑛
Since 𝐾 is 𝑓-simultaneously proximal, then there exists 𝑘0 ∈
𝐾 such that
𝑖=1
𝑖=1
𝑛
𝐹𝐾 (𝑥) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘) ≤∑ 𝑓 (𝑥𝑖 − 𝑘𝑚 ) 󳨀→ 0.
𝑖=1
(12)
(20)
𝑛
𝑖=1
≤∑ 𝑓 (𝑥𝑖 − 𝑘) .
𝑖=1
if and only if
𝑛
1
𝑓 (𝑥𝑖 − 𝑚󸀠 ) = 𝐹𝑀 (𝑥) , (13)
∑ 𝑓 (𝑥𝑖 − 𝑚0 ) = inf
∑
󸀠 ∈𝑀
𝛼
𝑚
𝑖=1
𝑛
𝑓
for all 𝛼 ∈ R − {0}, which implies that (1/𝛼)𝑚0 ∈ 𝑃𝑀(𝑥), so,
𝑓
𝑚0 ∈ 𝛼𝑃𝑀(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 𝑓simultaneously proximal, then 𝐾 is 𝑓-closed.
Proof . Since 𝑓 is a positive function, then ∑𝑛𝑖=1 𝑓(𝑥𝑖 ) ≥ 0 for
all 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 . Let {𝑘𝑚 } be a sequence of 𝐾 and
𝑓
Hence, 𝑘0 ∈ 𝑃𝐾 (𝑥).
̂𝐹 to be such that
For a subset 𝐾 of 𝑋, let us define 𝐾
𝑛
̂𝐹 = {𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 : 𝐹𝐾 (𝑥) = ∑𝑓 (𝑥𝑖 )} .
𝐾
𝑖=1
(21)
2
Example 12. Consider 𝑋 = (R2 ) and 𝐾 = {((𝑥1 , 𝑦1 ), (𝑥2 ,
𝑦2 )) : 𝑥𝑖 = 𝑦𝑖 , for all 𝑖 = 1, 2}. Let 𝑓(𝑥, 𝑦) = 𝑥2 + 𝑦2 ; then,
one can see that
̂𝐹 = {((𝑥1 , −𝑥1 ) , (𝑥2 , −𝑥2 ))} .
𝐾
(22)
4
Abstract and Applied Analysis
̂𝐹 , we prove the folUsing the previous definition of 𝐾
lowing theorem characterizing 𝑓-simultaneously proximal
subspaces.
Thus,
𝑛
∑ 𝑓 (𝑥𝑖 − 𝑘0 )
𝑖=1
Theorem 13. Let 𝐾 be a vector subspace of 𝑋. Then, 𝐾 is 𝑓̂𝐹 ,
simultaneously proximal in 𝑋 if and only if 𝑋𝑛 = 𝐷𝑘 + 𝐾
where 𝐷𝐾 = {(𝑘, 𝑘, . . . , 𝑘) : 𝑘 ∈ 𝐾}.
̂𝐹 . Then, for 𝑥 = (𝑥1 , 𝑥2 ,
Proof. Suppose that 𝑋𝑛 = 𝐷𝑘 + 𝐾
. . . , 𝑥𝑛 ) ∈ 𝑋𝑛 , there exists 𝑘1 = (𝑘0 , 𝑘0 , . . . , 𝑘0 ) ∈ 𝐷𝐾 and
̂𝐹 such that 𝑥 = 𝑦+𝑘1 . Hence, 𝑥−𝑘1 =
𝑦 = (𝑦1 , 𝑦2 , . . . , 𝑦𝑛 ) ∈ 𝐾
̂
𝑦 ∈ 𝐾𝐹 , and
𝑛
𝐹𝐾 (𝑦) = 𝐹𝐾 (𝑥 − 𝑘1 ) =∑ 𝑓 (𝑥𝑖 − 𝑘0 ) ,
(23)
𝑖=1
𝑛
= inf {∑ 𝑓 (𝑥𝑖 − 𝑘0 − 𝑘󸀠 ) : 𝑘󸀠 ∈ 𝐾} ,
𝑖=1
̂𝐹 .
which implies that 𝑥 − 𝑘0 ∈ 𝐾
̂𝐹 . Since 𝑓 is symmetric,
(2) Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝐾
then
𝑛
𝑛
𝑖=1
𝑖=1
𝑛
𝑛
= inf {∑ 𝑓 (𝑥𝑖 − 𝑘) : 𝑘 ∈ 𝐾}
∑ 𝑓 (𝑥𝑖 − 𝑘0 ) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘0 − 𝑘)
𝑘∈𝐾
𝑖=1
(27)
𝑖=1
∑ 𝑓 (−𝑥𝑖 ) =∑ 𝑓 (𝑥𝑖 )
and so
𝑛
𝑛
= inf {∑ 𝑓 (𝑥𝑖 − 𝑘0 + 𝑘0 − 𝑘) : 𝑘 ∈ 𝐾}
𝑖=1
𝑖=1
(24)
𝑛
(28)
𝑛
= inf {∑ 𝑓 (− (−𝑥𝑖 + 𝑘)) : −𝑘 ∈ 𝐾}
= inf
∑ 𝑓 (𝑥𝑖 − 𝑘󸀠 ) = 𝐹𝐾 (𝑥) .
󸀠
𝑖=1
𝑘 ∈𝐾 𝑖=1
𝑛
So, 𝐾 is 𝑓-simultaneously proximal.
Conversely, suppose that 𝐾 is 𝑓-simultaneously proximal
and 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 . Then, there exists 𝑘0 ∈ 𝐾 such
that
𝑛
= inf {∑ 𝑓 (−𝑥𝑖 + 𝑘) : −𝑘 ∈ 𝐾} .
𝑖=1
∑𝑛𝑖=1
𝑓(−𝑥𝑖 ) = inf{∑𝑛𝑖=1 𝑓(−𝑥𝑖 + 𝑘) : −𝑘 ∈ 𝐾},
Hence,
which implies that
̂𝐹 .
−𝑥 = (−𝑥1 , −𝑥2 , . . . , −𝑥𝑛 ) ∈ 𝐾
𝑛
∑ 𝑓 (𝑥𝑖 − 𝑘0 ) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘)
𝑘∈𝐾
𝑖=1
𝑖=1
𝑛
(25)
= inf ∑ 𝑓 (𝑥𝑖 − (𝑘󸀠 + 𝑘0 )) ,
𝑘∈𝐾
(3) Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ). Since 𝑥⊥𝐹 𝐾, then
𝑛
𝑛
𝑖=1
𝑖=1
∑ 𝑓 (𝑥𝑖 ) ≤ ∑𝑓 (𝑥𝑖 + 𝛼𝑘)
𝑖=1
(29)
∀𝛼 ∈ R, 𝑘 ∈ 𝐾.
𝑛
where 𝑘 = 𝑘󸀠 + 𝑘0 . If 𝑘1 = (𝑘0 , 𝑘0 , . . . , 𝑘0 ) ∈ 𝐷𝐾 , then
= ∑𝑓 (𝑥𝑖 − (−𝛼𝑘))
∀𝛼 ∈ R, 𝑘 ∈ 𝐾.
(30)
𝑖=1
𝑛
∑ 𝑓 (𝑥𝑖 − 𝑘0 ) = 𝐹𝐾 (𝑥 − 𝑘1 ) ,
𝑛
(26)
= ∑𝑓 (𝑥𝑖 − 𝑘󸀠 ) ,
𝑖=1
𝑘󸀠 ∈ 𝐾.
𝑖=1
̂𝐹 and 𝑋𝑛 = 𝐷𝑘 + 𝐾
̂𝐹 .
which implies that 𝑥 − 𝑘1 = 𝑘2 ∈ 𝐾
So,
Proposition 14. Let 𝑋 be a topological vector space and 𝐾𝑓simultaneous proximal subset of 𝑋. Then,
𝑓
̂𝐹 ;
(1) 𝑘0 ∈ 𝑃𝐾 (𝑥) if and only if 𝑥 − 𝑘0 ∈ 𝐾
(2) if 𝑓 is symmetric (i.e., 𝑓(−𝑥) = 𝑓(𝑥) for all 𝑥 ∈ 𝑋),
̂𝐹 if and only if −𝑥 ∈ 𝐾
̂𝐹 ;
then 𝑥 ∈ 𝐾
̂𝐹 , where 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 );
(3) if 𝑥⊥𝐹 𝐾, then 𝑥 ∈ 𝐾
̂𝐹 and 𝛼𝐾 = 𝐾, then 𝑥⊥𝐹 𝐾, where 𝑥 =
(4) if 𝑥 ∈ 𝐾
(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ).
𝑓
Proof. (1) Let 𝑘0 ∈ 𝑃𝐾 (𝑥) if and only if ∑𝑛𝑖=1 𝑓(𝑥𝑖 − 𝑘0 ) =
inf{∑𝑛𝑖=1 𝑓(𝑥𝑖 − 𝑘) : 𝑘 ∈ 𝐾}.
𝑛
𝑛
𝑖=1
𝑖=1
∑ 𝑓 (𝑥𝑖 ) ≤∑ 𝑓 (𝑥𝑖 − 𝑘󸀠 ) ,
𝑘󸀠 ∈ 𝐾.
(31)
̂𝐹 .
Hence, 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝐾
̂𝐹 and 𝛼𝐾 = 𝐾. Then,
(4) Let 𝑥 ∈ 𝐾
𝑛
𝑛
∑ 𝑓 (𝑥𝑖 ) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘)
𝑖=1
𝑘∈𝐾
𝑖=1
𝑛
= inf ∑ 𝑓 (𝑥𝑖 − 𝛼𝑘) ,
𝛼𝑘∈𝐾
since 𝛼𝐾 = 𝐾,
𝑖=1
𝑛
= inf ∑ 𝑓 (𝑥𝑖 + (−𝛼𝑘)) ,
𝑖=1
∀𝑘 ∈ 𝐾.
(32)
Abstract and Applied Analysis
5
Thus, for some 𝑚0 ∈ 𝑀, we have
Thus,
𝑛
𝑛
𝑖=1
𝑖=1
∑ 𝑓 (𝑥𝑖 ) ≤∑ 𝑓 (𝑥𝑖 + 𝛼󸀠 𝑘) ,
∀𝛼󸀠 ∈ R, ∀𝑘 ∈ 𝐾.
(33)
Hence, 𝑥⊥𝐹 𝐾.
̂𝐹 ) =
Theorem 15. Let 𝐾 be a vector subspace of 𝑋. If 𝜋(𝐾
𝑛
𝑋 /𝐷𝐾 , then 𝐾 is 𝑓-simultaneously proximal, where 𝜋 is the
canonical map 𝑥 → 𝑥 + 𝐷𝑘 .
̂𝐹 ) = 𝑋𝑛 /𝐷𝐾 and 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 .
Proof. Let 𝜋(𝐾
̂𝐹 . Hence, 𝑥−𝑦 = 𝑘0 for
Then, 𝑥+𝐷𝐾 = 𝑦+𝐷𝐾 for some 𝑦 ∈ 𝐾
̂
̂𝐹 +
some 𝑘0 ∈ 𝐷𝐾 . Thus, 𝑥 = 𝑦 + 𝑘0 ∈ 𝐾𝐹 + 𝐷𝑘 . Therefore, 𝐾
𝐷𝑘 = 𝑋𝑛 . By Theorem 15, 𝐾 is 𝑓-simultaneously proximal.
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
𝑓̃ : (𝑋/𝑀)𝑛 → R can be defined as follows:
𝑓̃ (𝑥1 + 𝑀, 𝑥2 + 𝑀, . . . , 𝑥𝑛 + 𝑀)
𝑛
(34)
𝑖=1
for each (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 .
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 , . . . , 𝑥𝑛 ) in 𝐾, then 𝑘0 +𝑀 is an 𝑓-best
simultaneous approximation to (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) + 𝑀 in 𝐾/𝑀.
̃
Proof. Suppose that 𝑘0 + 𝑀 is not 𝑓-best
simultaneous
approximation to (𝑥1 +𝑀, 𝑥2 +𝑀, . . . , 𝑥𝑛 +𝑀) in 𝐾/𝑀. Then,
𝑓̃ ((𝑥𝑖 − 𝑘0 +
𝑛
𝑀)𝑖=1 )
≰ 𝑓̃ ((𝑥𝑖 − 𝑘 +
𝑛
𝑀)𝑖=1 )
𝑖=1
𝑖=1
(39)
so,
𝑛
𝑛
𝑖=1
𝑖=1
∑ 𝑓 (𝑥𝑖 − (𝑘1 − 𝑚0 )) <∑ 𝑓 (𝑥𝑖 − 𝑘0 ) .
(40)
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.
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 , . . . , 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. 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 𝑓simultaneously 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 , . . . , 𝑥𝑛 ) + 𝑀 from
𝐾/𝑀, so,
𝑛
𝑛
𝑓̃ ((𝑥𝑖 − 𝑘0 + 𝑀)𝑖=1 ) ≤ 𝑓̃ ((𝑥𝑖 − 𝑘 + 𝑀)𝑖=1 ) ,
(35)
for at least 𝑘 ∈ 𝐾, say 𝑘1 ∈ 𝐾, such that
𝑛
𝑛
𝑓̃ ((𝑥𝑖 − 𝑘1 + 𝑀)𝑖=1 ) < 𝑓̃ ((𝑥𝑖 − 𝑘0 + 𝑀)𝑖=1 ) .
𝑛
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.
3. 𝑓-Simultaneous Approximation in
Quotient Space
= inf {∑ 𝑓 (𝑥𝑖 + 𝑦) : 𝑦 ∈ 𝑀}
𝑛
∑ 𝑓 (𝑥𝑖 − 𝑘1 + 𝑚0 ) <∑ 𝑓 (𝑥𝑖 − 𝑘0 ) ,
∀𝑘 ∈ 𝐾,
⇓
𝑛
(36)
𝑛
𝑓̃ ((𝑥𝑖 − 𝑘0 + 𝑀)𝑖=1 ) = inf ∑ 𝑓 (𝑥𝑖 − 𝑘0 + 𝑚)
𝑚∈𝑀
𝑖=1
𝑛
Since
≤ inf ∑ 𝑓 (𝑥𝑖 − 𝑘 + 𝑚) ,
𝑚∈𝑀
𝑛
𝑛
𝑓̃ ((𝑥𝑖 − 𝑘0 + 𝑀)𝑖=1 ) = inf {∑ 𝑓 (𝑥𝑖 − 𝑘0 + 𝑦) : 𝑦 ∈ 𝑀}
𝑖=1
(41)
𝑖=1
for all 𝑘 ∈ 𝐾. Note that
𝑛
≤∑ 𝑓 (𝑥𝑖 − 𝑘0 ) ,
𝑖=1
(37)
𝑛
inf ∑ 𝑓 (𝑥𝑖 − 𝑘0 + 𝑚)
𝑚∈𝑀
we have
𝑓̃ ((𝑥𝑖 − 𝑘1 +
𝑛
𝑀)𝑖=1 )
= 𝐹𝑀 (𝑥1 − 𝑘0 , 𝑥2 − 𝑘0 , . . . , 𝑥𝑛 − 𝑘0 )
𝑛
< ∑ 𝑓 (𝑥𝑖 − 𝑘0 ) .
𝑖=1
𝑖=1
(38)
≤ 𝐹𝑀 (𝑥1 − 𝑘, 𝑥2 − 𝑘, . . . , 𝑥𝑛 − 𝑘) .
(42)
6
Abstract and Applied Analysis
Since 𝑀 is 𝑓-simultaneously proximal in 𝑋, then there exists
𝑚0 ∈ 𝑀 such that
𝐹𝑀 (𝑥1 − 𝑘0 , 𝑥2 − 𝑘0 , . . . , 𝑥𝑛 − 𝑘0 )
𝑛
=∑ 𝑓 (𝑥𝑖 − 𝑘0 − 𝑚0 )
(43)
𝑖=1
𝑛
≤∑ 𝑓 (𝑥𝑖 − 𝑘 − 𝑚) ,
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
𝑓
ℓ1 ∈ 𝑃𝐾+𝑀(𝑥). Since 𝑀 ⊆ 𝐾 + 𝑀, we have that ℓ0 + 𝑀 and
𝑓
𝑓
ℓ1 + 𝑀 ∈ 𝑃(𝐾+𝑀)/𝑀(𝑥 + 𝑀) = 𝑃𝐾/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=1
∑ 𝑓 ((𝑥𝑖 − ℓ0 ) − 𝑚0 )
for all 𝑚 ∈ 𝑀 and 𝑘 ∈ 𝐾. So,
𝑛
𝑖=1
𝑛
𝑛
∑ 𝑓 (𝑥𝑖 − (𝑘0 + 𝑚0 )) ≤∑ 𝑓 (𝑥𝑖 − (𝑘 + 𝑚)) ,
𝑖=1
=∑ 𝑓 (𝑥𝑖 − ℓ1 )
(44)
𝑖=1
𝑖=1
for all 𝑚 ∈ 𝑀 and 𝑘 ∈ 𝐾. Hence,
𝑛
= inf {∑ 𝑓 (𝑥𝑖 − (ℓ0 + 𝑚))}
𝑚∈𝑀
𝑛
∑ 𝑓 (𝑥𝑖 − (𝑘0 + 𝑚0 ))
𝑖=1
𝑛
𝑖=1
≤ {∑ 𝑓 ((𝑥𝑖 − ℓ0 ) − 𝑚)} ,
(45)
𝑛
∀𝑚 ∈ 𝑀
𝑖=1
= inf {∑ 𝑓 (𝑥𝑖 − (𝑘 + 𝑚)) : 𝑚 ∈ 𝑀, 𝑘 ∈ 𝐾} .
= 𝐹𝑀 (𝑥 − ℓ0 ) .
𝑖=1
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. 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 𝑓simultaneously Chebyshev in 𝑋/𝑀.
Proof. Suppose not, then there exists (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) + 𝑀 ∈
𝑓̃
𝑋/𝑀, and 𝑘1 + 𝑀, 𝑘2 + 𝑀 ∈ 𝑃𝐾/𝑀((𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) + 𝑀) such
that 𝑘1 + 𝑀 ≠ 𝑘2 + 𝑀. Thus, 𝑘1 − 𝑘2 ∉ 𝑀. Since 𝑀 is 𝑓simultaneously proximal in 𝑋, then
𝑓
𝑃𝑀 (𝑥1 − 𝑘1 , 𝑥2 − 𝑘1 , . . . , 𝑥𝑛 − 𝑘1 ) ≠ 𝜙,
(46)
𝑓
𝑃𝑀 (𝑥1 − 𝑘2 , 𝑥2 − 𝑘2 , . . . , 𝑥𝑛 − 𝑘2 ) ≠ 𝜙.
𝑓
(47)
𝑓
Let 𝑚1 ∈ 𝑃𝑀(𝑥1 − 𝑘1 , 𝑥2 − 𝑘1 , . . . , 𝑥𝑛 − 𝑘1 ) and 𝑚2 ∈ 𝑃𝑀(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. 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
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 𝑓best simultaneous approximations 𝑚 and 𝑚󸀠 to 𝑥−𝑘 and 𝑥−𝑘󸀠
𝑓
from 𝑀, respectively. Therefore, we have 𝑚 ∈ 𝑃𝑀(𝑥 − 𝑘) and
𝑓
𝑚󸀠 ∈ 𝑃𝑀(𝑥 − 𝑘󸀠 ). Since 𝑀 ⊆ 𝐾 + 𝑀, 𝑘 + 𝑀 and 𝑘󸀠 + 𝑀 ∈
𝑓
𝑓
𝑃𝐾/𝑀(𝑥 + 𝑀) = 𝑃(𝐾+𝑀)/𝑀(𝑥 + 𝑀), so 𝑘 + 𝑚 and 𝑘󸀠 + 𝑚󸀠 ∈
𝑓
𝑃𝐾+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-quasisimultane𝑓
ously Chebyshev if 𝑃𝐾 (𝑥) is non-empty and 𝑓-compact set in
𝑋, for all 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 .
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 𝑋𝑛 /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 , . . . , 𝑥𝑛 ) ∈ 𝑋𝑛 and (𝑘𝑛 + 𝑀) a sequence in
𝑓̃
𝑃𝐾/𝑀(𝑥 + 𝑀). For every 𝑛, there exists 𝑚𝑛 ∈ 𝑀 such that
𝑓
𝑘𝑛 + 𝑚𝑛 = 𝑘𝑛󸀠 ∈ 𝑃𝐾 (𝑥). But since 𝑀 is a vector subspace, we
have
𝑘𝑛󸀠 + 𝑀 = 𝑘𝑛 + 𝑚𝑛 + 𝑀 = 𝑘𝑛 + 𝑀.
(48)
Abstract and Applied Analysis
7
Since 𝐾 is 𝑓-quasi-simultaneously Chebyshev of 𝑋, the
sequence {𝑘𝑛 } has a subsequence {𝑘𝑛𝑖 } which is 𝑓-convergent
𝑓
to 𝑘0 ∈ 𝑃𝐾 (𝑥), meaning that
𝑓 (𝑘𝑛𝑖 − 𝑘0 ) 󳨀→ 0.
(49)
𝑓̃ (𝑘𝑛𝑖 − 𝑘0 + 𝑀) ≤ 𝑓 (𝑘𝑛𝑖 − 𝑘0 ) 󳨀→ 0.
(50)
𝑓̃ (𝑘𝑛𝑖 − 𝑘0 + 𝑀) 󳨀→ 0.
(51)
But
Hence,
𝑓̂
̃
and 𝐾/𝑀
Consequently, 𝑃𝐾/𝑀(𝑥 + 𝑀) is 𝑓-compact
̃
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.
References
[1] W. W. Breckner and B. Brosowski, “Ein Kriterium zur Charakterisierung von sonnen,” Mathematica, vol. 13, pp. 181–188, 1971.
[2] T. D. Narang, “On 𝑓-best approximation in topological spaces,”
Archivum Mathematicum, vol. 21, no. 4, pp. 229–233, 1985.
[3] T. D. Narang, “Approximation relative to an ultra function,”
Archivum Mathematicum, vol. 22, no. 4, pp. 181–186, 1986.
[4] P. Govindarajulu and D. V. Pai, “On properties of sets related
to 𝑓-projections,” Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 457–465, 1980.
[5] D. V. Pai and P. Veermani, “Applications of fixed point theorems
in optimization and best approximation,” in Nonlinear Analysis
and Application, S. P. Singh and J. H. Barry, Eds., pp. 393–400,
Marcel Dekker, New York, NY, USA, 1982.
[6] M. R. Haddadi and J. Hamzenejad, “Best approximation in
TVS,” Caspian Journal of Mathematical Sciences, vol. 1, no. 2, pp.
75–70, 2012.
[7] S. V. R. Naidu, “On best simultaneous approximation,” Publications de l’Institut Mathématique, vol. 52(66), pp. 77–85, 1992.
[8] F. B. Saidi, D. Hussein, and R. Khalil, “Best simultaneous
approximation in 𝐿𝑝 (𝐼, 𝐸),” Journal of Approximation Theory,
vol. 116, no. 2, pp. 369–379, 2002.
[9] M. Abrishami Moghaddam, “On 𝑓-best approximation in
quotient topological vector spaces,” International Mathematical
Forum, vol. 5, no. 9–12, pp. 587–595, 2010.
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