ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII ”AL.I.CUZA” IAŞI
Tomul XLIX, s.I a, Matematică, 2003, f.2.
ON A MIN-MAX BEST APPROXIMATION PROBLEM IN A
LOCALLY CONVEX SPACE
BY
TEODOR PRECUPANU
Abstract. We consider the best approximation problem with respect to a family of
seminorms formulated as a min-max problem [13].
Since usual convexity conditions are too stringent to apply one of the classical minmax theorems we give a relaxated form for which we can use a general min-max theorem
established by Tanimoto [19]. Equivalently, a characterization of the best approximation
elements with respect to an associated norm (1.7) is obtained.
Let X be a linear space endowed with a separated family of seminorms
P= {pi ; i ∈ I}. Given a nonvoid subset A ⊂ X, the well-known problem of
best approximation of an element x ∈ X by elements of A can be formulated
in various forms. Taking into account the usual case of linear normed spaces,
a first possibility is to consider the best approximation problem with respect
to each seminorm of P ([7], [11], [17], [18]). But, in this situation the
problem can be reduced to a best approximation problem in linear normed
spaces considering the separated spaces associated to each seminorm of P.
In the same framework we may consider the best approximation with respect
to the Minkowsky functional of a convex circled absorbent set of X ([2], [11],
[16]) or the support functional of a convex circled set of dual space X ∗ ([11],
[20]). An interesting case is obtained when the best approximation problem
is formulated by properties concerning all seminorms of the family P. If
the locally convex space is metrizable the best approximation problem can
be formulated using cvasinorms or asymmetrical norms (see for instance
[1], [12], [21]). Other generalizations can be obtained using some special
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TEODOR PRECUPANU
2
seminorms (see [14], [15], [21]) or different functions defined in terms of
seminorms of P ([8], [18]).
1. In [13] was considered several types of the best approximation problems with respect to a family of seminorms for which was given several
relationships between optimality properties of those problems.
One of this problems was given in the following min-max form:
(1.1)
min sup p(x − y) = max inf p(x − y),
y∈A p∈P
p∈P y∈A
where x is a fixed element in X, i.e. the exists a pair (p0 , y0 ) ∈ P ×A such
that
(1.2)
p(x − y0 ) ≤ p0 (x − y0 ) ≤ p0 (x − y), for all (p, y) ∈ P × A.
Thus, the pair (p0 , y0 ) is a saddle point for the function (p, y) → p(x−y),
(p, y) ∈ P ×A. If we impose topological hypothesis for P (or I) and A and a
convexity property with respect to variables p(or i)and y, we can obtain the
existence criteria using min-max theorems. On the other hand, the problem
(1.1) can be decomposed in the following two dual optimization problems
(1.10 )
min sup p(x − y),
(1.100 )
max inf p(x − y).
y∈A p∈P
p∈P y∈A
It is well-known that the following inequality holds:
(1.3)
val(1.100 ) ≤ val(1.10 )
and (p0 , y0 ) is a saddle element if and only if y0 is an optimal solution of
(1.1’), p0 is an optimal solution of (1.1”) and (1.3) is just an equality.
Let us denote by
(1.4)
Pi (x; A) = {y0 ∈ X; pi (x − y0 ) ≤ pi (x − y), for all y ∈ A}
the set of all the best approximation elements of x ∈ X through the elements
of the set A with respect to the seminorm pi ∈ P. According to min-max
equality (1.1) it follows that a pair (i0 , y0 ) ∈ I × A is a solution if and only
if we have
(1.5)
pi0 (x − y0 ) = sup pi (x − y0 ),
i∈I
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ON A MIN-MAX APPROXIMATION IN A LOCALLY CONVEX SPACE
377
where
(1.6)
y0 ∈ Pi0 (x; A).
In the special case of a directed family of seminorms we obtain
Proposition 1.1. The min-max problem (1.1) has solutions if and only
if there exists i0 ∈ I such that equality (1.5) holds for at least an element
y0 ∈ Pi1 (x; A), where i1 ≥ i0 .
Proof. Indeed, if P is a directed family of seminorms then the equality
(1.5) holds for any i1 ≥ i0 and so (i1 , y0 ) is a saddle element whenever
y0 ∈ Pi1 (x; A).
Remark 1.1. In the special case of directed family of seminorms, if
(i0 , y0 ) ∈ I × A is a saddle element then the equality (1.5) becomes
pi0 (x − y0 ) = pi (x − y0 ) for all i ≥ i0 .
We return now to the problem (1.1’). It is obvious that this problem
is non-trivial if sup{pi (x − y); i ∈ I} < ∞ for at least one element y ∈ A.
Thus, if we denote
(1.7)
kuk∞ = sup{pi (u); i ∈ I}, u ∈ X,
and
(1.8)
X∞ = {u ∈ X; kuk∞ < ∞},
then the best approximation problem in min-max form can be considered
for a set A ⊂ X only for elements x in A + X∞ . Consequently, this problem
is propre for any element x ∈ X if and only if A + X∞ = X. A special
situation holds if X∞ = X when X becomes a normed space with respect
to the norm (1.7). In fact, the solutions of the problem (1.1’) are just the
best approximation elements of x ∈ X through elements of A with respect
to the norm (1.7). Thus, with the aid of the solutions for the min-max
problem (1.1) we can obtain characterizations of the best approximation
elements with respect to the norm k · k∞ in terms of seminorms of family
P.
Let us consider the following remarkable examples [13].
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TEODOR PRECUPANU
4
Example 1. Let (X, k · k) be a linear normed space, X ∗ its dual and
P = {px∗ ; x∗ ∈ X ∗ } the family of seminorms which generates the weak
topology, px∗ (x) = |x∗ (x)|, x ∈ X. It is obvious that X∞ = {0} and so
the problem (1.1) becomes trivial. But if, we consider the subfamily of
seminorms
∗
P0 = {px∗ ; x∗ ∈ S (0; 1)},
(1.9)
∗
where S (0; 1) = {x∗ ∈ X ∗ ; kx∗ k ≤ 1}, which also generates the weak
topology, we obtain X∞ = X and k · k∞ = k · k. Consequently, the solutions
of the problem (1.1’) coincide with the best approximation elements in
the linear normed space X. Moreover, according to the well-known dual
characterization of the best approximation elements (see for instance [3],
Corollary 3.1), if A is a closed convex set in X, an element y0 ∈ A is a best
approximation element of x ∈ X, through elements of A if and only if there
exists an element x∗0 ∈ X ∗ , kx∗0 k = 1 such that (x∗0 , y0 ) is a solution of minmax problem (1.1). Also, the min-max problem (1.1) has at least a solution
for any convex set A ⊂ X if the linear normed space X is reflexive, because
we can apply the usual min-max theorem for convex-concave functions an
compact domains considering the weak topology in X and weak∗ topology
in X ∗ ([3]). Therefore, with respect to the family of seminorms P0 , for a
nonvoid convex closed set A ⊂ X and an arbitrary fixed element x ∈ X\A,
the problem (1.1’) has a solution x ∈ A if and only if the problem (1.1”) has
a solution x∗0 such that (x∗0 , x) is a solution of the min-max problem (1.1).
Remark 1.2. The best approximation problem with respect to each
seminorm of a family (P) is invariable if we take a family of seminorms of
type P0 = {ki pi ; i ∈ I}, where (ki )i∈I is a family of strictly positive real
constants. But, the corresponding linear space X∞ can be modified. It is
easily to observe that for a given family of seminorms (P) there exists a
family of seminorms of type P0 such that its corresponding subspace X∞
coincides with the whole space X if and only if all seminorms of (P) are
continuous with respect to a certain norm of X.
Example 2. Let K be a compact and C(K) be the linear space of all
scalar continuous functions on K endowed with the pointwise convergence
topology generated by the following family of seminorms:
(1.10)
P = {pt ; t ∈ K}, where pt (x) = |x(t)|, x ∈ C(K).
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ON A MIN-MAX APPROXIMATION IN A LOCALLY CONVEX SPACE
379
In this case we also have X∞ = X and k · k∞ = k · kK , where k · kK is
the uniform convergence norm on K.
The problem (1.1’) becomes the best uniform approximation problem,
whereas the problem (1.1”) consists in finding an element t0 ∈ K such that
(1.11)
inf |x(t0 ) − y(t0 )| ≥ inf |x(t) − y(t)|, for all t ∈ K.
y∈A
y∈A
The existence of a such element t0 can be assured for sets A very special.
Moreover a characterization of the solutions x of the uniform best approximation problem by the existence of a t0 ∈ K, solution for (1.1”), such that
(t0 , x) to be a solution for (1.1) it is not possible even in simple cases.
To obtain the existence of the saddle elements via a usual min-max
theorem it is possible for a convex set A if we have a ”concavity” property
of the function t → |x(t) − y(t)|, t ∈ K, for every y ∈ A. But, a such
condition is usually very stringent.
Remark 1.3. The two examples are strongly connected because every linear normed space can be regarded as a linear subspace of a space
of continuous functions on a compact set; for this it is sufficient to take
K = S x∗ (0; 1) endowed with weak* topology on X ∗ . The essential distinction between the two presented examples consists in the fact that in the
examples 1 can be applied the min-max theorem under convexity and compactity hypothesis if the set A is closed and convex whereas in the example
2 the convexity-compactity conditions can be realized for y but they are
too stringent for t. An analogue situation appears in the case of the best
approximation with respect to a family of seminorms. Thus, it is possible to
use those min-max theorems which ask compacity and convexity conditions
with respect only to one of the variables.
In [19] Tanimoto established a such min-max result. In the case of
the linear normed space of vectorial continuous functions on a compact
set he obtained several important results specially in the case of finitedimensional sets. Certain results established by Tanimoto was reconsidered by Deutsch [5] for the space of continuous functions which vanish at
infinity, defined on a locally compact, space using directly a characterization theorem for the best approximation elements obtained independently
by Deutsch and Maserick [6] and Havinson [9].
2. Taking into account the generality of the min-max theorem of Tanimoto we can establish some existence results for the best approximation
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TEODOR PRECUPANU
problem with respect to a family of seminorms P under ”weakened” minmax form.
Thus we can obtain characterizations of the best approximation elements
with respect to the norm k k∞ with the aid of the values of finite systems
of seminorms from the family P.
Theorem A[19]. Let U be a nonvoid convex compact set in a separated
linear topological space and let I be an arbitrary nonvoid set. Let us consider
the function F : U × I → R such that F (·, i) is convex and lower semicontinuous in U for every i ∈ I. Then
(2.1)
min sup F (u, i) = lim
u∈U i∈I
n→∞
sup
(λn ,in )∈I n
min
u∈U
n
X
λk F (u, ik )
k=1
where
I n = {(λn , in ) ∈ Rn × I n ; λn = (λ1 , λ2 , · · · , λn ), λk ≥ 0, k = 1, n,
(2.2)
n
X
λk = 1, in = (i1 , i2 , · · · , in )}, n = 1, 2, · · ·
k=1
Theorem 2.1. Let A be a nonvoid convex weakly compact set in the
locally convex space (X, τP ). Then, for every element x ∈ A + X∞ we have
(2.3)
min sup pi (x − y) = lim
y∈A i∈I
n→∞
sup
(λn ,in )∈I n
min
y∈A
n
X
λk pik (x − y).
k=1
Proof. Let us take U = A and let us consider the weak topology on X.
It is obviously that the function y → pi (x − y) is convex and τp -continuous
on X, for each i ∈ I. Consequently this function is also weakly continuous
and so we can apply theorem A taking F (u, i) = pi (x − u), (u, i) ∈ A × I.
It is clear that the elements of A + X∞ have the best approximation
elements for the norm k k∞ with respect to the convex weakly compact set
A. According to the theorem 2.1 these best approximation elements satisfy
the equality (2.3).
In fact, this equality characterizes the best approximation elements.
Theorem 2.2. An element y0 ∈ A is a best approximation of the
element x ∈ A + X∞ through elements of A with respect to the norm k k∞
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ON A MIN-MAX APPROXIMATION IN A LOCALLY CONVEX SPACE
381
if and only if
(2.4)
|x − y0 |∞ = lim
n→∞
min
sup
y∈A
(λn ,in )∈I n
n
X
λk pik (x − y).
k=1
Proof. Indeed, from (2.4) it follows that for every ε > 0 there exist
n
P
λεk = 1 and iε1 , iε2 , · · · , iεn ∈ I such that
λε1 , λε2 , · · · , λεn ≥ 0 with
k
|x − y0 |∞ < ε + min
(2.5)
y∈A
n
X
λεk piεk (x − y).
k=1
On the other hand, we have
|x − ye|∞ ≥
n
X
λεk piεk (x − ye) ≥ min
y∈A
k=1
n
X
λεk piεk (x − y)
k=1
for all ye ∈ A. Consequently we obtain |x − ye|∞ > |x − y0 |∞ − ε, for every
ye ∈ A and ε > 0, i.e. |x − ye|∞ ≥ |x − y0 |∞ for all ye ∈ A, as claimed.
Conversely, if y0 ∈ A is a best approximation element we have
|x − y0 |∞ ≤ |x − y|∞ = sup pi (x − y), for all y ∈ A.
i∈I
Therefore, for every ε > 0 and y ∈ A there exists iε,y ∈ A such that
|x − y0 |∞ − ε < piε,y (x − y). But it is obvious that
pi (x − y) ≤
n
X
sup
(λ̄k ,īk )∈I¯k
λk pik (x − y) for each i ∈ I and y ∈ A.
k=1
Hence
|x − y0 |∞ − ε <
sup
min
(λn ,in )∈I k
y∈A
n
X
λk pik (x − y), ε > 0, n ∈ N,
k=1
i.e.
(∗)
|x − y0 |∞ ≤ lim
n→∞
sup
(λn ,in )∈I k
min
y∈A
n
X
k=1
λk pik (x − y),
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TEODOR PRECUPANU
8
which assures the equality (2.4) because the reverse inequality is always
true.
Remark 2.1. The equality (2.4) is equivalent to the inequality (2.5)
for some (λ̄εn , īεn ) ∈ I¯n , where ε > 0 is arbitrary.
The equality (2.1), and consequently (2.3), can be considered as a ”minn
P
λk F (u, ik ), where h = (n, ln , in )
sup” equality for the function Fe(u, h) =
∈
∞
S
k=1
({n} × I n ) because the sequence from right side of (2.1) is nondecreas-
n=1
ing. Thus, we have the possibility to obtain similar results using other
”min-sup” theorems.
Remark 2.2. If the family of the seminorms P is directed, then (2.3)
and (2.4) are even min-sup equalities.
Indeed, if (2.5) holds, then there exists an i∗ ∈ I such that pik ,ε (x − y) ≤
pi∗ (x − y) for all y ∈ A and consequently.
kx − y0 k∞ < ε + min pi∗ (x − y).
y∈A
Hence, if we renounce to the convex combinations in the right side of
the equality (2.3), respectively (2.4), the value of the ”min-sup” does not
change.
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Received: 10.IV.2003
Faculty of Mathematics,
University “Al. I. Cuza”,
11, Carol I,
700506 – Iaşi,
ROMANIA