Almirag.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 5, No. 1, 2004

pp. 49- 70

A topological approach to Best Approximation
Theory

Samuel G. Moreno, Jose Maŕıa Almira, Esther M.

Garćıa–Caballero and J. M. Quesada∗

Abstract. The main goal of this paper is to put some light in several
arguments that have been used through the time in many contexts of
Best Approximation Theory to produce proximinality results. In all
these works, the main idea was to prove that the sets we are considering
have certain properties which are very near to the compactness in the
usual sense. In the paper we introduce a concept (the wrapping) that
allow us to unify all these results in a whole theory, where certain ideas
from Topology are essential. Moreover, we do not only cover many of
the known classical results but also prove some new results. Hence we
prove that exists a strong interaction between General Topology and
Best Approximation Theory.

2000 AMS Classification: 41A65, 54-99

Keywords: Wrapping, Best Approximation, Proximinality, Compactness.

1. Introduction

One of the central problems in Best Approximation Theory can be roughly
formulated in the following way. Let X be a set, A a nonempty subset of X
and x ∈ X. If there exists a real valued function d on X × X that provides
a notion of gap between points in X, we want to know about the existence of
points a ∈ A such that

d(x, a) = inf
b∈A

d(x, b).

The points a ∈ A satisfying the previous relation are called the best approxi-
mations to x from A. The subset A is said to be proximinal if for all x ∈ X
there exists a best approximation to x from A.

∗Research supported by Junta de Andalućıa, Grupos FQM0178 and FQM0268.



50 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

The proximinality of a subset A can be assured if there exists a topology for
X large enough to be the functions d(x, ·) : A −→ R sequentially lower semi-
continuous and small enough in order to be the domain A countably compact.
The main tools used in the known proofs of the results concerning the existence
of best approximations can be clearly distinguished in the following cases.

(a): If X is a normed space and the gap function is the metric derived
from the norm, the Functional Analysis arguments will give a common
thread of compactness. As a sample of the most notorious results of
this kind we give the following list.

• Nonempty closed subsets of finite dimensional subspaces of normed
linear spaces are proximinal.

• Nonempty weakly closed subsets of a reflexive Banach space (in
particular, any nonempty closed convex subset) are proximinal.

• Nonempty weakly star closed subsets of the dual space of a normed
space are proximinal.

(b): If X is not normed, or X is normed but the gap function d is not
a metric. In this situation the set X is usually a function space and
the techniques used in the proofs will depend on each particular case.
The arguments will derive, in general, from Measure Theory and they
will provide the convergence of certain sequences of functions. As an
example we mention that the subset of non decreasing functions in
the Orlicz space Lφ(0, 1) is proximinal in the sense that for each func-
tion f ∈ Lφ(0, 1) there is a non decreasing function g ∈ Lφ(0, 1) that
minimizes the gap function

d(f, ·) =

∫ 1

0

φ(|f − ·|)dµ

over the subset of non decreasing functions in Lφ(0, 1).
Another interesting situation is when X is quasi-metric space (see

for example [21], where a characterization of best approximation result
is proved in this context).

The main purpose of this paper is to give a common thread of compactness in
the proofs of proximinality. To achieve our aim we introduce and extend some
well known facts of Best Approximation Theory to general topological spaces
with some structure. Our description covers and extends some classical situ-
ations in normed spaces (see [6]) and in metric spaces. Moreover some recent
developments in topological vector spaces are covered through our method.

The present paper is organized as follows. In Section 2 we state the main
definitions. We use the following idea: let (X, ρ) be a metric space, x ∈ X and
A a nonempty subset of X. The set of best approximations to x from A is
defined by

P(x, A) = {a ∈ A : ρ(x, a) = ρ(x, A)},



Topological approach to Best Approximation 51

where ρ(x, A) = inf
a∈A

ρ(x, a). If B(x, r) denotes the closed ball of center x and

radius r, we have

P(x, A) = A ∩ B(x, ρ(x, A)) = A ∩





⋂

r>ρ(x,A)

B(x, r)



 .

This description gives the idea to extend some notions of Best Approxi-
mation Theory to more general spaces. In order to estimate the gap between
points in some space we will introduce the concept of wrapping as an increasing
family of closed sets with nonempty interior whose union is the whole space.
Section 3 is devoted to the application of the framework introduced in the pre-
vious section to Metric Spaces, Topological Vector Spaces and Function Spaces.
We cover some of the classical situations and some recent descriptions due to
different authors. In Section 4 we show that our description gives a common
thread of compactness for the proofs of proximinality in some of the most in-
teresting and well known examples. Finally, we also characterize proximinality
in terms of countable compactness. The situation described here in order to
introduce the notion of wrapping has been applied by other authors not only
in Approximation Theory but also in Selection theory (in Michael’s approach,
see [15]) and in Fixed Point Theory (see [12]).

2. Definitions and Preliminaries Results

Let (X, τ) be a Hausdorff topological space and let us denote by C(X) the
family of closed subsets of X. A wrapping for X is a function ξ : X× [0, ∞) −→
C(X) such that, for each x ∈ X, satisfies:

(i) ξ(x, 0) = {x},
(ii) for all r > 0 there is an open set θ(x, r) with x ∈ θ(x, r) ⊂ ξ(x, r),
(iii) for all r, s ≥ 0, if r ≤ s then ξ(x, r) ⊆ ξ(x, s),
(iv) for all s ≥ 0,

⋃

r>s

ξ(x, r) = X.

The set ξ(x, r) is called the ξ-ball of center x and radius r.
If ξ is a function from X × [0, ∞) into the power set P(X) that fulfills the

axioms (iii) and (iv) of the above definition, then it is called a pre–wrapping
for X.

Fixed a wrapping ξ for X, A a nonempty subset of X and x ∈ X, we define:

• the radius set of the ξ-balls of center x with nonempty intersection with
A

(2.1) Fξ(x, A) = {r ≥ 0 : A ∩ ξ(x, r) 6= ∅},

• the ξ-distance of x to A

(2.2) dξ(x, A) = inf Fξ(x, A),



52 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

• the set of best ξ-approximations to x from A

(2.3) Pξ(x, A) = A
⋂





⋂

r∈Fξ(x,A)

ξ(x, r)



 .

The subset A will be called ξ-proximinal if Pξ(x, A) is nonempty for all x ∈ X,
and it will be called ξ-Chebyshev if there is a function pξ(·, A) : X −→ A such
that Pξ(x, A) = {pξ(x, A)} for all x ∈ X (i.e. if Pξ(x, A) is a singleton for all
x ∈ X).

By means of the previous definitions we have the following statements:

(i) Pξ(·, A) is a function from X to P(A),
(ii) x ∈ A if and only if Pξ(x, A) = {x},
(iii) the set Fξ(x, A) is either the interval (dξ(x, A), ∞) or [dξ(x, A), ∞).

Moreover, in the last case we have Pξ(x, A) = A
⋂

ξ(x, dξ(x, A)) 6= ∅,
(iv) dξ(x, A) = inf

a∈A
dξ(x, {a}) = inf

a∈A
inf{r ≥ 0 : a ∈ ξ(x, r)},

(v) {y ∈ X : dξ(x, {y}) ≤ s} =
⋂

r>s

ξ(x, r), for each x ∈ X,

(vi) dξ(·, A) is a function from X to [0, ∞) and dξ(·, A)|A = 0.

Associated to the concept of wrapping, we would like to introduce the fol-
lowing properties: Given ξ a wrapping for X, we say that it satisfies the

Intersection property: If for all x ∈ X and s ≥ 0, ξ(x, s) =
⋂

r>s

ξ(x, r).

Triangular property: If for all x ∈ X, r ≥ 0, y ∈ ξ(x, r) and s ≥ 0,
ξ(y, s) ⊂ ξ(x, r + s).

A-Adherence property: If for all x ∈ X, if A
⋂

(

⋂

r>0

ξ(x, r)

)

6= ∅ then

x ∈ A.

If ξ satisfies the intersection property it is clear that

(2.4) Pξ(x, A) = A ∩ ξ(x, dξ(x, A)) = {a ∈ A : dξ(x, {a}) = dξ(x, A)},

and so Pξ(x, A) 6= ∅ if and only if Fξ(x, A) = [dξ(x, A), ∞).
In case that ξ satisfies the triangular property, and x, y, z ∈ X, we have that

for each ε > 0,

y ∈ ξ(x, dξ(x, {y}) +
ε

2
) and z ∈ ξ(y, dξ(y, {z}) +

ε

2
),

so z ∈ ξ(x, dξ(x, {y}) + dξ(y, {z}) + ε) for all ε > 0. Therefore

(2.5) dξ(x, {z}) ≤ dξ(x, {y}) + dξ(y, {z})

and hence the function ρ : X × X −→ [0, ∞) defined by

ρ(x, y) = dξ(x, {y}) + dξ(y, {x})



Topological approach to Best Approximation 53

is a pseudometric in X. Reciprocally, if (2.5) is fulfilled and ξ satisfies the
intersection property, it also satisfies the triangular property.

Without difficulty it can be shown that {x} =
⋂

r>0

ξ(x, r) for all x ∈ X, if

and only if for all A ⊂ X the wrapping ξ has the A-adherence property.
We can characterize the closed subsets in A by means of the proximinality

and adherence properties as follows

Proposition 2.1. Let (X, τ) be a Hausdorff regular space. A nonempty subset
A of X is closed if and only if there exists a wrapping ξ for X with the A-
adherence property such that A is ξ-proximinal.

Note 1. A Hausdorff space is regular if for each point x and each closed subset
A, if x 6∈ A, then there are disjoint open sets U and V such that x ∈ U and
A ⊂ V.

Proof Let A be a closed subset of X. If A = X we can choose the wrapping
defined by ξ(x, 0) = {x} and ξ(x, r) = X for r > 0. If A is a proper subset
of X and x ∈ X \ A there are disjoint open sets Ux, Vx such that x ∈ Ux and
A ⊂ Vx; we define the wrapping ξ by:

ξ(x, r) =







{x} if r = 0,
X \ Vx if 0 < r < 1,
X if r ≥ 1,

if x 6∈ A, and ξ(x, 0) = {x} and ξ(x, r) = X for r > 0 in other case. Then,
in case that x ∈ A we have Pξ(x, A) = {x} and if x 6∈ A it follows that
Pξ(x, A) = A. Consequently A is ξ-proximinal, and it is straightforward to
verify that ξ has the A-adherence property.

To show the converse suppose that A is not closed and let x ∈ A \ A. For
every wrapping ξ, since the open sets θ(x, r) have nonvoid intersection with

A, we have that Pξ(x, A) = A
⋂

(

⋂

r>0

ξ(x, r)

)

. So if A is ξ-proximinal, the

wrapping cannot have the A-adherence property. 2

Note 2. The previous result is in the same line as Theorem 6 of [11].

Let ρ be a metric in a set X. For each x ∈ X, the function ρx : X −→ [0, ∞)
defined by ρx(y) = ρ(y, x) gives the distance to x. By definition, the metric
topology for X is the smallest one containing the sets

S(x, r) := {y ∈ X : ρ(y, x) < r} = ρ−1x ([0, r)),

for all x ∈ X and r ≥ 0. Hence the metric topology for X is the smallest one
that makes the functions ρx continuous at x. If A ⊂ X, the uniform continuity
of the function ρ(·, A) follows from the symmetry and the triangle inequality
of the metric ρ. With a similar pattern we can state the following result.



54 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

Proposition 2.2. Let ξ be a wrapping for X and ∅ 6= A ⊂ X. If ξ satisfies
the triangular property and dξ(·, {x}) is continuous at x ∈ X, then dξ(·, A) is
continuous at x.

Proof Let {xi}i∈I be a net in X such that xi −→ x. Given ε > 0, there is a i1 ∈
I such that xi ∈ ξ(x,

ε
2
) for i ≥ i1. For each i ∈ I, let ai ∈ A ∩ ξ(xi, dξ(xi, A) +

ε
2
). Since ξ satisfies the triangular property, then ai ∈ ξ(x, dξ(xi, A) + ε) for

i ≥ i1, and hence

(2.6) dξ(x, A) − dξ(xi, A) ≤ ε.

To prove the other inequality, first observe that x ∈ ξ(xi, dξ(xi, {x}) +
ε
4
) for

all i ∈ I and let aε ∈ A ∩ ξ(x, dξ(x, A) +
ε
4
). Since aε ∈ ξ(xi, dξ(xi, {x}) +

dξ(x, A) +
ε
2
), then dξ(xi, A) − dξ(x, A) ≤ dξ(xi, {x}) +

ε
2
. Finally, notice that

there exists i2 ∈ I such that

(2.7) dξ(xi, A) − dξ(x, A) ≤ ε,

for all i ≥ i2, since dξ(·, {x}) is continuous at x.
If i0 is an upper bound of {i1, i2}, it follows from (2.6) and (2.7) that

|dξ(xi, A) − dξ(x, A)| ≤ ε,

for all i ≥ i0.2

3. Metric Spaces, Topological Vector Spaces and Function
Spaces.

3.1. Metric Spaces. Let (X, ρ) be a metric space, and let ∅ 6= A ⊂ X. In
the sequel, the closed ball of center x and radius r will be denoted by B(x, r).
The function ξ : X × [0, ∞) −→ C(X) defined by

ξ(x, r) = B(x, r)

is the “natural” wrapping for X, and it clearly satisfies the intersection and
the triangular properties. We recall that

P(x, A) = {a ∈ A : ρ(x, a) = ρ(x, A)} = A ∩ B(x, ρ(x, A))

where ρ(x, A) is the distance of x to A. We can generalize this wrapping by
merely introducing a real function f with suitable properties and considering
the balls B(x, f(r)). In some sense, the best approximation problem with this
new kind of wrapping will be show to be equivalent to the standard problem
in the space (X, ρf ), where ρf is a metric derived from ρ and f.

More precisely, a function f : [0, ∞) −→ [0, ∞) is said to be of type A (abbr.
TA) if

(i) f(r) = 0 if and only if r = 0,
(ii) f is right continuous (lim

r↓s
f(r) = f(s)),

(iii) f is superadditive (f(r) + f(s) ≤ f(r + s)).



Topological approach to Best Approximation 55

Let f be a TA function and 0 ≤ r < s. We have f(r) < f(r)+f(s−r) ≤ f(s),
thus f is strictly increasing. Moreover, nf(1) ≤ f(n) for each nonnegative
integer n, so that lim

r→∞
f(r) = ∞.

Let us now give some natural examples of TA functions: The first one is
given by f(r) = r + [r] (where [r] denotes the greatest integer smaller or equal
to r) is a TA function. On the other hand, if f : [0, ∞) −→ [0, ∞) is a convex
function that vanishes only at 0, then it is an strictly increasing continuous TA
function: if p ∈ [0, ∞), then

(3.8) f(λp) ≤ λf(p) + (1 − λ)f(0) = λf(p),

for all λ ∈ [0, 1]. Therefore, if 0 < r ≤ s and λ ∈ (0, 1], we have

1

λ
f(λ(r + s)) ≤ f(r + s).

Taking λ =
s

r + s
and using (3.8), we get

f(r) + f(s) ≤
r

s
f(s) + f(s) ≤ f(r + s).

Let (X, ρ) be a metric space, and let f be a TA function. By the properties
of f, it is clear that the function ξf , defined on X × [0, ∞) by

ξf (x, r) = B(x, f(r))

is a wrapping for X, that satisfies the intersection and the triangular properties.
Moreover, we can state the following result.

Proposition 3.1. Let (X, ρ) be a metric space, A a nonempty subset of X and
f a TA function. For every x ∈ X, we have

Pξf (x, A) = A∩B(x, f(dξf (x, A))) where dξf (x, A) = inf{r ≥ 0 : f(r) ≥ ρ(x, A)}.

Proof The first relation follows from the intersection property of the wrapping
ξf . Clearly,

{r ≥ 0 : f(r) > ρ(x, A)} ⊂ {r ≥ 0 : A∩B(x, f(r)) 6= ∅} ⊂ {r ≥ 0 : f(r) ≥ ρ(x, A)},

so that we have an inverted chain of inequalities of infimums of such sets. Now f
is right continuous and strictly increasing, so that inf{r ≥ 0 : f(r) > ρ(x, A)} =
inf{r ≥ 0 : f(r) ≥ ρ(x, A)} and the second statement of the proposition holds.2

Let f be a TA function, and let the right inverse of f be defined by

f−1ex (x) = inf{r ≥ 0 : f(r) ≥ x} = inf f
−1([x, ∞)),

for x ≥ 0. We can prove now the following result.

Proposition 3.2. Let f−1ex be the right inverse of a TA function f. Then
f−1ex is a function from [0, ∞) onto [0, ∞) which is non decreasing, continuous,
subadditive ( f−1ex (x + y) ≤ f

−1
ex (x) + f

−1
ex (y) ) and vanishes only at 0.



56 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

Proof Since lim
r→∞

f(r) = ∞, then f−1([x, ∞)) is nonempty (and bounded

below) for each x ≥ 0. Under the hypothesis on f (strictly increasing and right
continuous), for each x ≥ 0 there exists s ≥ 0 such that f−1([x, ∞)) = [s, ∞),
hence f−1ex (x) = min f

−1([x, ∞)).
Let r ≥ 0 and x = f(r). It is clear that f−1ex (x) = r = f

−1(x), hence f−1ex is
a surjective extension of the inverse function f−1 : f([0, ∞)) −→ [0, ∞). Also,
if 0 ≤ x ≤ y, then [y, ∞) ⊂ [x, ∞) and

f−1ex (x) = min f
−1([x, ∞)) ≤ min f−1([y, ∞)) = f−1ex (y),

so that f−1ex is non decreasing.
Let x > 0 and let a = lim

y↓x
f−1ex (y) and b = lim

y↑x
f−1ex (y). By the surjec-

tivity of f−1ex , it follows that a = b. In the same way we get that 0 =
f−1ex (0) = lim

y↓0
f−1ex (y). Hence f

−1
ex is continuous in [0, ∞). The condition

f−1ex (x) = 0 if and only if x = 0, is a direct consequence of the right continuity
of f.

Finally, let x, y > 0. There exist r, s > 0 such that

f(t) < x ≤ f(r) for t ∈ [0, r), and f(t) < y ≤ f(s) for t ∈ [0, s),

so that x + y ≤ f(r) + f(s) ≤ f(r + s), and f−1ex (x + y) ≤ f
−1
ex (f(r + s)) =

r + s = f−1ex (x) + f
−1
ex (y). Hence f

−1
ex is a subadditive function.2

Let ρ be a metric for X and let f be a TA function. One can easily verify
that the composition

ρf = f
−1
ex ◦ ρ

is a metric in X. In fact, ρ and ρf are equivalent metrics. Moreover, if Bσ
denotes the closed ball in the metric space (X, σ), we have that

(3.9) Bρ(x, f(r)) = Bρf (x, r),

which follows from

Bρ(x, f(r)) = {y ∈ X : ρ(y, x) ≤ f(r)} ⊂ {y ∈ X : f
−1
ex (ρ(y, x)) ≤ r} = Bρf (x, r),

and from

Bρf (x, r) = {y ∈ X : f
−1
ex (ρ(y, x)) ≤ r} ⊂ {y ∈ X : ρ(y, x) ≤ f(r)} = Bρ(x, f(r)),

(the last inclusion is due to ρ(x, y) ≤ f(f−1ex (ρ(x, y)))).
We can also establish the following relation

dξf (x, A) = inf{r ≥ 0 : f(r) ≥ ρ(x, A)} = f
−1
ex (ρ(x, A))

= f−1ex ( inf
a∈A

ρ(x, a)) = inf
a∈A

f−1ex (ρ(x, a)) = inf
a∈A

ρf (x, a)

= ρf (x, A).(3.10)

In view of (3.9) and (3.10), Proposition 3.1 takes the form



Topological approach to Best Approximation 57

Proposition 3.3. Let (X, ρ) be a metric space, A a nonempty subset of X and
f a TA function. For every x ∈ X, we have

Pξf (x, A) = A ∩ Bρf (x, ρf (x, A)).

Hence the best approximation problem in the metric space (X, ρ) with the wra-
pping ξf (x, r) = Bρ(x, f(r)), is equivalent to the approximation problem in
(X, ρf) with the usual wrapping ξ(x, r) = Bρf (x, r).

3.2. Topological Vector Spaces. In [17] the following problem has been
considered. Let X be a separated locally convex space and let f : X −→ R be
a continuous convex function satisfying f(0) = 0. If A is a nonempty closed
subset of X, the authors define the number

fA(x) = inf{f(x − a) : a ∈ A},

and the so-called f–projection set

Pf,A(x) = {a ∈ A : f(x − a) = fA(x)}.

They define properties of A related to the set valued mapping Pf,A(·) and
explore several relationships between these properties and the continuity of
this mapping. Taking into account that for each r > 0 the sub-level set

Cr = {x ∈ X : f(x) ≤ r}

is a closed convex absorbing set containing 0 in its interior, is immediate to
verify that the function ξ defined by

ξ(x, r) =

{

{x} if r = 0,
x − Cr if r > 0,

is a wrapping for X that satisfies the intersection property. Of course we have
dξ(x, A) = fA(x) and Pξ(x, A) = Pf,A(x), hence this minimization problem
can be described and studied with the tools we have introduced in this paper.

Also the following problem can be found in [2]. Let C be a closed bounded
convex subset of a Banach space X which has the origin as an interior point and
let fC denote the Minkowski functional with respect to C. Given a nonempty
closed bounded subset A ⊂ X and a point x ∈ X, we consider the minimization
problem which consists in proving the existence of a point a0 ∈ A such that

fC(a0 − x) = inf{fC(a − x) : a ∈ A}.

In this subsection we are going to introduce a wrapping in a topological
vector space in order to cover and extend the problem above.

Let (X, τ) be a (Hausdorff) topological vector space, and let C be a convex
neighborhood of 0. For all x ∈ X, it is easy to check that

(i) for each r > 0, the set x+rC is an convex neighborhood of x contained
in the closed convex set x + rC,

(ii) if 0 ≤ r ≤ s, then x + rC ⊂ x + sC,



58 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

(iii) for each y ∈ X, there exists t ≥ 0 such that y belongs to x + tC, and
so

⋃

r≥0

(x + rC) = X.

In the light of the above properties, we can state the following result (we
remind the reader that a halfline in X is a set of the form {x + ty : t ≥ 0},
where x, y ∈ X and y 6= 0).

Proposition 3.4. Let (X, τ) be a Hausdorff topological vector space with a
convex neighborhood of the origin, C, whose closure does not contain halflines.
The function ξC : X × [0, ∞) −→ C(X) defined by

ξC(x, r) = x + rC

is a wrapping for X that satisfies the intersection and the triangular properties.

Proof We will prove the following two statements:

(1): For each x ∈ X, r ≥ 0, and for each y ∈ x + rC and s ≥ 0,

y + sC ⊂ x + (r + s)C.

(2): For each x ∈ X and s ≥ 0,

x + sC =
⋂

r>s

(x + rC).

As the first step for the proof of (1) we shall prove that for all r, s ≥ 0,

(3.11) rC + sC = (r + s)C.

Let x, y ∈ C and let r, s be positive. By convexity of C, it follows that

1

r + s
(rx + sy) =

r

r + s
x +

s

r + s
y ∈ C,

which implies that rC + sC ⊂ (r + s)C. The reverse inclusion is trivial.
Now, let r, s > 0 and let y ∈ x + rC. If z belongs to y + sC, then, using

(3.11), we have z ∈ x + rC + sC = x + (r + s)C. This concludes the proof of
(1).

The proof of (2) goes as follows. Since trivially

x + sC ⊂
⋂

r>s

(x + rC),

we have to prove the reverse inclusion. First consider s > 0, and let y ∈
⋂

r>s

(x + rC). If {rn} is a sequence of numbers greater than s and converging

to s, we have that

C ∋
1

rn
(y − x) →

1

s
(y − x),

and hence y ∈ x + sC. This implies
⋂

r>s

(x + rC) ⊂ x + sC



Topological approach to Best Approximation 59

for each s > 0. Finally, let y ∈
⋂

r>0
(x + rC). By

1

r
(y − x) ∈ C

for each r > 0, if y 6= x, then the halfline {t(y − x) : t ≥ 0} is contained in C,
which is absurd. So

⋂

r>0

(x + rC) ⊂ {x},

and this concludes the proof.2

Observe that an open set which does not contain halflines needs not to be
bounded. Consider, for example, the set

C = {{xn} ∈ l1 : |xn| < n for each n},

where l1 denotes the space of sequences {xn}
∞
n=1 of real numbers which are

absolutely summable. It is clear that C is a convex set containing 0. Moreover,
let x = {xn} ∈ C and define

εx = min
n

{n − |xn|}.

If y ∈ {z ∈ l1 : ‖z − x‖1 < εx}, then, for each n,

|yn| − |xn| ≤ |yn − xn| ≤ ‖y − x‖1 < εx ≤ n − |xn|,

and therefore C is open. It is straightforward to verify that the convex set

C = {{xn} ∈ l1 : |xn| ≤ n for each n}

does not contain halflines and, however, it is unbounded (if xn = {(n −
1)δmn}

∞
m=1, then xn ∈ C and sup

n
‖xn‖1 = ∞).

Let C be a convex subset of a Hausdorff Topological Vector Space, with 0
as an interior point and which contains no halflines. The Minkowski functional
of C, defined by

fC(x) = inf{r > 0 :
x

r
∈ C},

is a non-negative, positive homogeneous, convex, continuous and subadditive
function, that vanishes only at 0. Moreover, the convex sets

C1 = {x ∈ X : fC(x) < 1} and C2 = {x ∈ X : fC(x) ≤ 1}

(open and closed, respectively) satisfy

C1 ⊂ C ⊂ C ⊂ C2.

The Minkowski functional provides the following characterization of the set
of best ξC-approximations.



60 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

Proposition 3.5. Let (X, τ) be a Hausdorff topological vector space with a
convex neighborhood of the origin, C, whose closure does not contain halflines,
and let A be a nonempty subset of X. For each x ∈ X, we have

PξC (x, A) = A ∩
(

x + dξC (x, A)(C \ C)
)

(3.12)

= {a ∈ A : f
C
(a − x) = dξC (x, A)},

where dξC (x, A) = inf
a∈A

f
C
(a − x).

Proof It follows from (2.4) that PξC (x, A) = A ∩
(

x + dξC (x, A)C
)

.
We first consider the case dξC (x, A) > 0. For a ∈ PξC (x, A), we will show

that a 6∈ x+dξC (x, A)C. Suppose this is not so. If {tn} is a sequence of positive
numbers such that tn → 0, then (1 + tn)a − tnx → a, and hence there exists
tm > 0 and c ∈ C such that (1 + tm)a − tmx = x + dξC (x, A)c, so that

a = x +
dξC (x, A)

1 + tm
c ∈ x +

dξC (x, A)

1 + tm
C.

Hence we have dξC (x, A) ≤
dξC (x, A)

1 + tm
, an absurd. This shows that

A ∩
(

x + dξC (x, A)C
)

⊂ A ∩
(

x + dξC (x, A)(C \ C)
)

.

The reverse inclusion and the case dξC (x, A) = 0 are obvious. Thus we have
proved that

PξC (x, A) = A ∩
(

x + dξC (x, A)(C \ C)
)

.

The proof ends by noting that

dξC (x, {a}) = inf{r ≥ 0 : a ∈ x + rC} = inf{r ≥ 0 :
a − x

r
∈ C} = f

C
(a − x).

2

We have also the following characterization of the ξC-distance of x to A.

Proposition 3.6. If

∅ 6= A ∩
(

x + rC
)

⊂ x + r(C \ C)

for some r ≥ 0, then r = dξC (x, A) and consequently PξC (x, A) is nonempty.

Proof It suffices to consider r > 0. First observe that d = dξC (x, A) ≤ r.
Supposing d < r, we get

∅ 6= A ∩

(

x +
d + r

2
C

)

⊂ A ∩
(

x + rC
)

⊂ x + r(C \ C).

Thus there exist c1 ∈ C and c2 ∈ C \ C, such that

d + r

2
c1 = rc2.



Topological approach to Best Approximation 61

Let t = d+r
2r

∈ (0, 1). Since c1 +
1−t
−t

C is a neighborhood of c1, there exists

c3 ∈ C
⋂

(

c1 +
1−t
−t

C
)

. Therefore

c2 = tc1 = tc3 + t(c1 − c3) ∈ tC + (1 − t)C ⊂ C.

This contradicts that c2 6∈ C, and we conclude d = r. 2

The set of best ξC-approximations to x from A inherits the invariance pro-
perties of the space X, and it is easy to

verify that for s > 0, t ≥ 0 and y ∈ X, we have

Pξ(sC) (tx − y, tA − y) = tPξC (x, A) − y.

A convex subset U of a topological vector space will be called strictly convex
provided that the relations x, y ∈ U \ int(U) and x 6= y imply x+y

2
∈ int(U).

Note that if U is a convex neighborhood of the origin, then int(U) = U and U is
strictly convex if U is so. It is possible to establish uniqueness as a consequence
of strict convexity of the set that generates the wrapping.(For normed linear
spaces see Lemma 3.2. of [19])

Proposition 3.7. Let (X, τ) be a Hausdorff topological vector space with a
convex neighborhood of the origin, C, whose closure does not contain halflines,
and let A be a nonempty, convex and ξC-proximinal subset of X. If C is
strictly convex, then A is ξC-Chebyshev. Moreover, if for each convex and ξC-
proximinal subset A of X we have that A is ξC-Chebyshev, then C is strictly
convex.

Proof First consider that C is strictly convex. Let a1, a2 ∈ PξC (x, A). In case

a1 6= a2, we have d = dξC (x, A) > 0, and so
ai−x

d
∈ C \ C. By the assumption

on C, we have
a1+a2

2
− x

d
∈ C.

Since A is convex, this means that a1+a2
2

∈ A ∩ (x + dC), which contradicts
(3.12). This proves the first claim.

Now suppose that C is not strictly convex. Then there exist distinct points
a1, a2 ∈ C \ C such that

a1+a2
2

∈ C \ C. Let the mapping g from [0, 1] into X
be defined by

g(t) = (1 − t)a1 + ta2.

Since g is continuous and [0, 1] is compact, the convex set A = g([0, 1]) is
compact. From (3.12) and the fact that f

C
(· − x) is continuous, it follows that

A is ξC-proximinal. Moreover, since fC(a) = 1 = dξC (0, A) for each a ∈ A (this
is not difficult to check), then A = PξC (0, A). Hence A is not ξC-Chebyshev.
This proves the second claim.2

For a ∈ A, Sa will denote the set of all points in X having a as a best
ξC-approximation, i.e.

Sa = {x ∈ X : a ∈ PξC (x, A)} = {x ∈ X : fC(a − x) = dξC (x, A)}.



62 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

Clearly Sa is nonempty (a ∈ Sa). Let {xd}d∈D be a net in Sa which converges to
x. Then f

C
(a−xd) = dξC (xd, A). Since fC(a−·) and dξC (·, A) are continuous,

we have
f
C
(a − x) = dξC (x, A),

hence x ∈ Sa and Sa is closed.
It is interesting to note that if we assume that A is convex we can prove a

geometrical condition of the sets Sa. This condition has been established, for
normed linear spaces, in [19].

Proposition 3.8. Let (X, τ) be a Hausdorff topological vector space with a
convex neighborhood of the origin, C, whose closure does not contain halflines,
and let A be a nonempty and convex subset of X. Then for each a ∈ A, the set
Sa is a cone with vertex a.

Proof If x ∈ Sa, and we denote d = dξC (x, A), then there exists c1 ∈ C \ C
such that a = x + dc1. We shall prove that for each t ≥ 0

(3.13) ∅ 6= A ∩
(

(1 − t)a + tx + tdC
)

⊂ (1 − t)a + tx + td(C \ C).

Suppose this is not so. Then for some b ∈ A and some c2 ∈ C, we would have

b = (1−t)a+tx+tdc2 = (1−t)x+(1−t)dc1 +tx+tdc2 = x+d((1−t)c1 +tc2).

In case that t ∈ (0, 1), by the fact that (1 − t)c1 + tc2 belongs to C, we
have b ∈ x + dC, an absurd. If t ∈ (1, ∞), using that A is convex, we get
1
t
b + (1 − 1

t
)a = 1

t
x + d((1

t
− 1)c1 + c2) + (1 −

1
t
)x + (1 − 1

t
)dc1 = x + dc2 ∈ A,

again a contradiction.
From (3.13) and proposition 3.6 , we deduce

dξC ((1 − t)a + tx, A) = tdξC (x, A),

for all non-negative t. Therefore

f
C
(a − ((1 − t)a + tx)) = tf

C
(a − x) = tdξC (x, A) = dξC ((1 − t)a + tx, A).

Whence (1 − t)a + tx belongs to Sa.2

Without the hypothesis of convexity on A, we can establish the following
corollaries of the above result.

Corollary 3.9. Let (X, τ) be a Hausdorff topological vector space with a convex
neighborhood of the origin, C, whose closure does not contain halflines, and let
∅ 6= A ⊂ X. If a ∈ A and x ∈ Sa, then (1 − t)a + tx ∈ Sa for each t ∈ [0, 1].

Corollary 3.10. Let (X, τ) be a Hausdorff topological vector space with a
convex neighborhood of the origin, C, whose closure does not contain halflines,
and let A be a nonempty and ξC-Chebyshev subset of X. Then

pξC ((1 − t)pξC (x, A) + tx, A) = pξC (x, A),

for each x ∈ X and t ∈ [0, 1].

The previous two statements are well known in the context of normed linear
spaces (see [24]).



Topological approach to Best Approximation 63

3.3. Function Spaces. We recall that a convex function φ from [0, ∞) into
[0, ∞) is said to be ∆2-convex at 0 if there exists K > 0 such that

(3.14) φ(2x) ≤ K φ(x),

for each x ≥ 0.
If φ : [0, ∞) −→ [0, ∞) is a ∆2-convex function at 0, with φ(0) = 0 and

φ 6≡ 0 then it follows rather easily that

(a) For all x > 0, φ(x) > 0,
(b) φ is superadditive ,
(c) φ is strictly increasing,
(d) There exists M ≥ 1 such that φ(x+y) ≤ M(φ(x)+φ(y)) for all x, y ≥ 0.

Let (Ω, A, µ) be a finite measure space, φ : [0, ∞) −→ [0, ∞) a ∆2-convex
function at 0 and let Lφ(Ω) := Lφ(Ω, A, µ) denote the class of the µ-equivalent

measurable functions, f : Ω −→ R, such that
∫

Ω

φ(|f|)dµ < ∞,

where we assume that f = g if µ{ω ∈ Ω : f(ω) 6= g(ω)} = 0. It is well known
that Lφ(Ω) is a real linear space.

For r ≥ 0, f ∈ Lφ(Ω), we define the sets of functions,

θ(f, r) =

{

g ∈ Lφ(Ω) :

∫

Ω

φ(|f − g|)dµ < r

}

and

ξ(f, r) =

{

g ∈ Lφ(Ω) :

∫

Ω

φ(|f − g|)dµ ≤ r

}

.

In order to define a topology in Lφ(Ω) we consider the family τφ of subsets of
Lφ(Ω),

(3.15) τφ = {O ⊂ Lφ(Ω) : ∀f ∈ O, ∃r > 0 such that θ(f, r) ⊂ O}

Proposition 3.11. The following conditions hold true

(a) (Lφ(Ω), τφ) is a Hausdorff topological space.
(b) {θ(f, r) : f ∈ Lφ(Ω), r > 0} is a basis for τφ.
(c) For all f ∈ Lφ(Ω) and r ≥ 0, ξ(f, r) is τφ–closed.

Proof
(a) By definition (3.15), ∅ and Lφ(Ω) belong to τφ. Moreover, if {Oλ}λ∈Λ is
an arbitrary family of sets in τφ, is immediate that ∪λ∈ΛOλ ∈ τφ. Finally, let
Oi, i = 1, 2, · · · , n, be a finite family of sets in τφ. For all f ∈ ∩

n
i=1Oi, there are

ri > 0 such that θ(f, ri) ⊂ Oi, i = 1, 2, · · · , n. Taking r0 = min
1≤i≤n

ri, we have

θ(f, r0) =

n
⋂

i=1

θ(f, ri) ⊂

n
⋂

i=1

Oi,

and therefore ∩ni=1Oi ∈ τφ. This proves that τφ is a topology in Lφ(Ω).



64 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

Now let f, g ∈ Lφ(Ω), f 6= g and consider

α =

∫

Ω

φ(|f − g|)dµ > 0.

In order to prove that (Lφ(Ω), τφ) is a Hausdorff topological space, we will show
that

(3.16) θ (f, α/(2M)) ∩ θ (g, α/(2M)) = ∅.

If (3.16) is false, then there exists h ∈ Lφ(Ω), such that h ∈ θ (f, α/(2M)) ∩
θ (f, α/(2M)). But this is not possible since

∫

Ω

φ(|f − g|)dµ ≤

∫

Ω

φ(|f − h| + |g − h|)dµ

≤ M

(
∫

Ω

φ(|f − h|)dµ +

∫

Ω

φ(|g − h|)dµ

)

< α.

(b) By definition of τφ it suffices to show that θ(f, r) belongs to τφ. We will
prove that for all g ∈ θ(f, r) there exists s > 0 such that θ(g, s) ⊂ θ(f, r).
Suppose the contrary. Then there exists a sequence {gn} in Lφ(Ω), such that

∫

Ω

φ(|g − gn)|)dµ <
1

2n
and

∫

Ω

φ(|f − gn|)dµ ≥ r.

By the Jensen inequality

φ

(

1

µ(Ω)

∫

Ω

|g − gn|dµ

)

≤
1

µ(Ω)

∫

Ω

φ(|g − gn|)dµ,

we obtain
∫

Ω

|g − gn|dµ ≤ µ(Ω)φ
−1

(

1

2nµ(Ω)

)

,

and therefore ‖g − gn‖L1(Ω) → 0. Then there exists a subsequence {gnk}
such that gnk → g, µ-almost everywhere in Ω. Since φ is continuous, then
φ(|f − gnk |) → φ(|f − g|), µ-a.e. On the other hand,

φ(|f − gnk|) ≤ M (φ(|f − g|) + φ(|g − gnk|)) ≤ M (φ(|f − g|) + h) ,

where h =
∞
∑

k=1

φ(|g − gnk|). Since, φ(|g − gnk|) ≥ 0 on Ω, then

∫

Ω

hdµ =

∞
∑

k=1

∫

Ω

φ(|g − gnk|)dµ <

∞
∑

k=1

1

2nk
< ∞

and therefore M (φ(|f − g|) + h) ∈ L1(Ω). Finally, applying the Lebesgue
Dominated Convergence Theorem, we get

∫

Ω

φ(|f − g|)dµ = lim
k→∞

∫

Ω

φ(|f − gnk|)dµ ≥ r,

and we obtain a contradiction.
(c) If we consider the complement of ξ(f, r), the proof follows the same pattern
of (b).2



Topological approach to Best Approximation 65

Then we easily deduce the following

Corollary 3.12. The family {ξ(f, r) : f ∈ Lφ(Ω), r ≥ 0} is a wrapping for
(Lφ(Ω), τφ).

4. ξ-Proximinality

In this section we prove a simple yet general proximinality result. It is
general because it includes, as special cases, some of the most interesting and
well known examples of proximinality. Is simple because the proof requires
nothing but the definition of countable compactness.

We recall that a Hausdorff topological space is called countably compact if
every countable open covering has a finite subcovering. Countable compactness
admits a Heine-Borel type argument: a Hausdorff topological space is count-
ably compact if and only if every family of closed subsets having the finite
intersection property also has the countable intersection property. As a con-
sequence, each descending sequence of nonempty closed subsets has nonempty
intersection. Finally, let us remember that a space is countably compact if and
only if every sequence of points of the space has an accumulation point.

The following result generalizes a result by Singer ([24], p. 383).

Theorem 4.1. Let ξ be a pre–wrapping for (X, τ) and let A be a nonempty
subset of X. If τ′ is a Hausdorff topology in X such that for each x ∈ X there
exists a non increasing sequence {εn} of positive numbers tending to 0 such
that

(i) A ∩ ξ(x, dξ(x, A) + ε1) is τ
′-countably compact,

(ii) A ∩ ξ(x, dξ(x, A) + εn) is τ
′-closed for n > 1,

then A is ξ-proximinal.

Proof If for each x ∈ X we define, for n ≥ 1,

An = A ∩ ξ(x, dξ(x, A) + εn),

then {An+1} is a non increasing sequence of nonempty and τ
′-closed subsets

of A1. Since A1 is τ
′-countably compact,

(4.17) ∅ 6=
⋂

n≥1

An+1 = A ∩
⋂

{ξ(x, r) : r > dξ(x, A)}.

If we assume Fξ(x, A) = (dξ(x, A), ∞), then by 4.17 we have that Pξ(x, A) is
nonempty. The same conclusion follows trivially in case Fξ(x, A) = [dξ(x, A), ∞).
Thus A is ξ-proximinal.2

It may be surprising that some of the most famous existence results in
normed linear spaces and also results in function spaces that are not normed
can be obtained as consequences of the result above. The following list of
examples is intended to be a representative sampling of this fact.

A: Nonempty closed subsets of finite dimensional subspaces of normed
linear spaces are proximinal.



66 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

We consider the “natural” wrapping for the normed linear space, i. e. ξ(x, r) =
B(x, r), where B(x, r) stands for the closed ball of center x and radius r. If A
is a closed subset of a finite dimensional subspace, then the sets A ∩ B(x, r)
are bounded closed subsets contained in a finite dimensional space, hence they
are compact.

This example is emblematic because it gives an affirmative answer, with
an elegant formulation, to the problem that gave rise to Best Approximation
Theory, namely, the possibility on finding, within the algebraic polynomials
of degree least or equal to a fixed n, the nearest one (in the sense of uniform
norm) to some continuous fixed function.

This result was proved by F. Riesz [20] in 1918, although the corresponding
result in the setting of polynomial approximation was proved by Chebyshev in
1859 (see [3], [4] and [25]).

Note 3. For the following example, we recall first that σ(X, X∗) stands for
the weak topology on X, i. e., the topology defined by the family of seminorms
{px∗ : x

∗ ∈ X∗}, where px∗(x) = |x
∗(x)| for each x ∈ X. On the other hand,

σ(X∗, X) stands for the weak star topology on X∗, i. e., the topology defined
by the family of seminorms {px : x ∈ X}, where px(x

∗) = |x∗(x)| for each
x∗ ∈ X∗.

B: Every nonempty σ(X, X∗)–closed subset of a reflexive Banach space
X is proximinal.

We consider again the natural wrapping for X, i. e. ξ(x, r) denotes the closed
ball of center x and radius r. If X is reflexive (i.e., the natural embedding of X
into its double dual, X∗∗, is surjective), then the unit ball B(0, 1) is σ(X, X∗)–
compact (in fact, this is a characterization of reflexivity, see [9]). The functions
fx and gα (with x ∈ X and α 6= 0 ), defined by

fx(y) = x + y and gα(y) = αy for all y ∈ X

are homeomorphisms in (X, σ(X, X∗)). Then, as the continuous image of a
compact set is compact, we have that

B(x, r) = x + rB(0, 1) = fx(gr(B(0, 1)))

is σ(X, X∗)-compact for all x ∈ X and all r > 0. So if A is a nonempty and
σ(X, X∗)–closed subset of a reflexive space, then the sets

A ∩ B(x, r)

are weakly compact. Thus, A is ξ-proximinal.
Mazur proved (see [14]) that the norm closure of a convex subset equals

its weak closure. Then the nonempty closed convex subsets of a reflexive Ba-
nach space are proximinal. This appears firstly in a paper of Day (see [5]) in
1941. In fact, he gives this result for Banach spaces with a weakly compact
unit ball. Day used the results of Milman and Alaoglu and Birkhoff which
gave the same result for uniformly convex spaces. By other way, Smulian (see
[23]) gave a characterization of the reflexivity of Banach spaces in terms that
every descending sequence of nonempty closed, bounded and convex subsets



Topological approach to Best Approximation 67

had nonvoid intersection. The previous example can be deduced from this
characterization.

C: Nonempty and σ(X∗, X)-closed subsets of the dual space X∗ of a
normed space X are proximinal.

For x∗ ∈ X∗ and r ≥ 0, the closed balls (in the usual norm of the dual spaces)
of center x∗ and radius r are used to define the wrapping. More precisely,

ξ(x∗, r) = B∗(x∗, r) = {y∗ ∈ X∗ : ‖y∗ − x∗‖ ≤ r}

where ‖x∗‖ = sup {|x∗(x)| : x ∈ B(0, 1)}.
The Alaoglu theorem states that B∗(0∗, 1) is σ(X∗, X)-compact. The balls

B∗(x∗, r) are then weakly star compact. If A is a nonempty and σ(X∗, X)-
closed subset of X∗, then the sets

A ∩ B∗(x∗, r)

are σ(X∗, X)-compact. Thus A is ξ-proximinal.
The previous example, for the case of linear subspaces, appears firstly in a

paper of Hirschfeld ([10], 1958) with a wrong proof. Later, in 1960, it appeared
in a paper of Phelps (see [18])

D: The subset of non decreasing functions in L∞(0, 1) is proximinal.

Let µ be the Lebesgue measure on the interval (0, 1). Recall that an extended
real valued Lebesgue measurable function f on (0, 1) is said to be essentially
bounded if there exists some real number a ≥ 0 such that µ({x ∈ (0, 1) :
|f(x)| > a}) = 0. If f is essentially bounded then the essential supremum of f
is defined by

‖f‖∞ = inf{a ≥ 0 : µ({x ∈ (0, 1) : |f(x)| > a}) = 0}.

Let L∞(0, 1) denote the set of all essentially bounded Lebesgue measurable
functions on (0, 1), two functions being identified if they differ only on a set of
measure zero, and let A ⊂ L∞(0, 1) be the subset of non decreasing functions
from (0, 1) into R. Under pointwise linear operations, (L∞(0, 1), ‖ · ‖∞) is a
real Banach space, and each equivalence class in L∞(0, 1) contains a bounded
function.

It is clear that the function ξ defined by

ξ(f, r) = B(f, r) = {g ∈ L∞(0, 1) : ‖g − f‖∞ ≤ r}

where f ∈ L∞(0, 1) and r ≥ 0, is a wrapping for (L∞(0, 1), ‖ · ‖∞).
Using a bounded function as a representative of each equivalence class f ∈

L∞(0, 1) we have, for r > 0,

(4.18) B(0, r) ⊂
∏

x∈(0,1)

[−r, r].

With the aid of Tychonoff theorem we can state that
∏

x∈(0,1)

[−r, r] is compact

in the cartesian product topology and without effort we can prove that the
nonempty set A∩B(f, ‖f‖∞) is closed in the product topology. The remainder



68 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

of the proof is therefore devoted to showing that the set A∩B(f, ‖f‖∞) is com-
pact in the product topology. But, taking into account the previous comments,
this a direct consequence of (4.18) and the fact that B(f, ‖f‖∞) ⊂ B(0, 2‖f‖∞).

E: The subset of non decreasing functions in Lφ(0, 1) is φ-proximinal,
i.e. for each function f ∈ Lφ(0, 1) there is a non decreasing function
g ∈ Lφ(0, 1) such that

∫ 1

0

φ(|f − g|)dµ ≤

∫ 1

0

φ(|f − h|)dµ

for each non decreasing function h ∈ Lφ(0, 1).

Let us consider the wrapping ξ described in Subsection 3.3. Then, for a fixed

f ∈ Lφ(0, 1) and for r ≥ 0, we have ξ(f, r) =
{

g ∈ Lφ(0, 1) :
∫ 1

0
φ(|f − g|)dµ ≤ r

}

.

We shall denote, for shortness, d = dξ(f, A) and Aε = A ∩ ξ(f, d + ε), where ε
is any positive number. We will show that A is ξ-proximinal by proving that
for all ε > 0, every sequence in Aε has an accumulation point in the topology
τφ (hence Aε is τφ-countably compact), and that Aε is τφ-closed.

First consider a sequence {gn} in Aε. Then using the Jensen inequality we
get

∫ 1

0

|f − gn|dµ ≤ φ
−1

(
∫ 1

0

φ(|f − gn|)dµ

)

,

and therefore ‖gn‖L1 ≤ K ‖f‖L1 + φ
−1(d + ε). By the previous inequality it

is straightforward to verify that the functions gn are uniformly bounded in
each closed subinterval [a, b] ⊂ (0, 1). Applying the Helly theorem we get a
subsequence {gnk} such that gnk → g, µ-almost everywhere in (0, 1). Since φ
is continuous, then φ(|f − gnk|) → φ(|f − g|), µ-a.e. The function g is non
decreasing and by Fatou Lemma we have

(4.19)

∫ 1

0

φ(|f − g|)dµ ≤ d + ε.

On the other hand, by (4.19) and the fact that φ(|g|) ≤ M (φ(|f − g|) + φ(|f|)),
we can assure that g ∈ Lφ(0, 1) and therefore g ∈ Aε.

Since |g − gnk| ≤ |f − g| + |f − gnk|, then

φ(|g − gnk|) ≤ M (φ(|f − g|) + φ(|f − gnk|)) ∈ L1(0, 1).

Applying the Lebesgue Dominated Convergence Theorem, we have

lim
k→∞

∫ 1

0

φ(|g − gnk|)dµ = 0.

Thus g is an accumulation point (in the τφ topology) of {gn}.
The sets Aε are τφ-closed since they are τφ-countably compact and τφ is first

countable.
To close this section, we characterize, for proper closed subsets of regular

spaces, the proximinality property in terms of countable compactness. This
result is in the same spirit as Theorem 5 of [11]



Topological approach to Best Approximation 69

Proposition 4.2. Let (X, τ) be a Hausdorff regular space. A proper and closed
subset A of X is countably compact if and only if it is ξ-proximinal for any
wrapping ξ.

Proof Let us consider x ∈ X \ A. The regularity of X implies that there are
open sets Ox, OA such that x ∈ Ox, A ⊂ OA and Ox ∩ OA = ∅. Hence x is
an interior point of the closed set UA = X \ OA. Now suppose that A is not
countably compact and let {Fn} be a sequence of nonempty and relative closed
subsets of A such that Fn+1 ⊂ Fn and ∩

∞
n=1Fn = ∅. Since A is closed, the

relative closed subsets Fn are closed.
For 0 < r ≤ 1, let nr denotes the integer such that

1

2nr
< r ≤

1

2nr−1
.

We define the wrapping ξ in X by

ξ(x, r) =







{x} if r = 0,
UA ∪ Fnr if 0 < r ≤ 1,
X if r > 1,

and, for y 6= x, ξ(y, 0) = {y} and ξ(y, r) = X for r > 0. Thus, we have

Pξ(x, A) = A ∩

(

∞
⋂

n=1

(UA ∪ Fn)

)

= ∅.

This implies that A is not proximinal with respect to the wrapping described
above.2

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70 S. G. Moreno, J. M. Almira and E. M. Garćıa-Caballero and J. M. Quesada

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Received October 2002

Accepted February 2003

S. G. Moreno, J. M. Almira, E. M. Garćıa-Caballero. (samuel@ujaen.es,
jmalmira@ujaen.es, emgarcia@ujaen.es)
Departamento de Matemáticas, Universidad de Jaén, E.U.P. Linares, 23700
Linares (Jaén), Spain.

J. M. Quesada (jquesada@ujaen.es)
Departamento de Matemáticas, Universidad de Jaén, E.P.S. Jaén. Avda. de
Madrid 35, 23071 Jaén, Spain.