Set Convergence and the Class of Compact Subsets

ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ ”AL.I.CUZA” IAS¸I Tomul XLVII, s.I a, Matematic˘a,2001, f.2.

SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS

G. APREUTESEI

1. Introduction. If (X, d) is a metric space and Cl(X) is the class of closed subsets of X, one can defined on Cl(X) some topologies or convergences, called hypertopologies or hyperconvergences. In recent years, many efforts have been done in order to find some conditions concerning the space (X, d) which assure the coincidence of some known hypertopologies or hyperconvergences on the fixed class Cl(X) ([7], [6], [11] etc.); if now the space (X, d) is fixed, an other problem, initiated in [1], is to specify some subclasses (even ”the best” subclass) A of Cl(X) on which two precised hypetopologies (or hyperconvergences) coincide. In [1], the author establish that on the class K(X) of the compact sets of X, many hypertopologies coincide, more precisely

K(X) P ≡ V ≡ `f ≡ H ⇓ bP ≡ AW where the topologies V, lf, H, P, AW, bP will be described in the Section 2. In this paper, we offer new results concerning the class K(X) with respect to Pompeiu-Hausdorff and Wijsman convergences for monotone sequences (Section 3, Theorem 3.1), as well as Mosco and Pompeiu-Hausdorff convergences of nets (Section 4, Theorem 4.1, which is formulated in terms of necessary and sufficient condition). 264 G. APREUTESEI 2 2. Preliminaries and notations. Let (X, d) be a metric space. A distance ρ is called equivqlently to d if they induce the same topology on X. We denote by D the family of all metrics that are equivalent to d. Also we give the notations:

B(X) = {A ∈ Cl(X); A is d-bounded };

S(a, ε) = {x ∈ X; d(a, x) < ε} with a ∈ X, ε > 0 –the ball with center a and radius ε; B(a, ε) = {x ∈ X; d(a, x) ≤ ε} with a ∈ X, ε > 0 –the closed ball with center a and radius ε; [ Sε(A) = S(a, ε), ε–enlargement of the set A; a∈A cl A = the d-closure of the set A. In order to deﬁne the desired convergences, we introduce: i) the Hausdorﬀ excess of A with respect to B

e(A, B) = sup{d(a, B); a ∈ A};

ii) the r-excess of A with respect to B

er(A, B) = e(A ∩ B(x0, r),B), where x0 is ﬁxed in X and r is a positive real number. iii) for a net (Ai)i∈I of closed subsets of X (where I is a nonvoid directed index set), the lower limit , respectively the upper limit of (Ai) are deﬁned as follows

lim inf(Ai) = {a ∈ X; ∀i ∈ I ∃ ai ∈ Ai such that a = lim ai}

= {a ∈ X; lim d(a, Ai) = 0}, respectively

lim sup(Ai) = {a ∈ X; ∃ J ⊂ I coﬁnally in I such that

∀j ∈ J ∃ aj ∈ Aj with a = lim aj}   \ [ = cl  Ai . i∈I j≥i If X is a linear topological space, we denote the topological dual of X by X∗ and the weak topology on X by w. 3 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 265

Also, C(X) = {A ∈ Cl(X); A is convex} and Ac the intersection of C(X) with A. The support function of a closed convex set A is

s(x∗,A) = sup{x∗(a); a∈A}, where x∗ ∈X∗. In many cases, it will be convenient to consider ([17]) a hyperconver- gence c as the conjunction of a ”lower convergence” (c−) and an ”upper convergence” (c+). We consider X a metric space. The Pompeiu-Hausdorﬀ convergence is H, the conjunction of H− and H+, where (H−) lim e(A, Ai) = 0,

(H+) lim e(Ai,A) = 0.

If denoted by H(A, B) = max{e(A, B), e(B,A)}, then (Ai) is H - convergent to A iﬀ lim H(Ai,A) = 0. The Attouch-Wets convergence is AW , the conjunction of AW− and AW+, where (AW−) lim er(A, Ai) = 0,

(AW+) lim er(Ai,A) = 0, for every positive real r and arbitrary x0. The Wijsman convergence is W , the conjunction of W− and W+, where (W−) ∀x∈X, d(x, A) ≥ lim sup d(x, Ai),

(W+) ∀x∈X, d(x, A) ≤ lim inf d(x, Ai).

The Kuratowski convergence is K, the conjunction of K− and K+, where (K−) A ⊂ lim inf(Ai),

(K+) lim sup(Ai) ⊂ A. 266 G. APREUTESEI 4

We notice that K− is equivalently to W−. The proximal convergence is P , the conjunction of H+ and K−. The b-proximal convergence is bP , the conjunction of AW+ and K−. The locally ﬁnite convergence is lf, the supremum of all convergences H(d), when d ∈ D ([7]). The Vietoris convergence is V , the supremum of all convergences P (d), when d ∈ D ([7]). The Mosco convergence is M,the conjunction of K− and M+ (see [12], [2]), where (M+) w − lim sup(Ai) ⊂ A, and the linear convergence is L, the conjunction of K− and S+, where

∗ ∗ ∗ ∗ (S+) ∀x ∈X , lim sup s(x ,Ai) ≤ s(x ,A). We note that all this convergence, except the Kuratowski and Mosco convergences, are topological. The relations between they are given on K(X) by the table of ﬁrst page. On Cl(X) or C(X) we have also the implications ([16] and [17]):

P =⇒ L =⇒ M =⇒ K;

AW =⇒ W =⇒ K. Finally, we recall that a family A of closed sets of a metric space (X, d) is stable with respect to closed subsets if for any set A ∈ A and B ⊂ A with B ∈ Cl(X) it follows that B ∈ A.

The Wijsmann convergence for the monotone sequence. In the following, we intent to observe if the convergences (or the topologies) Wijsman and Hausdorﬀ agree with on K(X). This is not true, as we observe from the followind example:

Example 3.1. Lets consider `1 with the usual norm. ∗ We take An = {x ∈ `1; x = ten, t ∈ [0, 1]}, ∀n ∈ IN , where by en we denote the unit vector of the canonical base in `1, and A = {0}. Obviously, ∗ the sets A and An, n ∈ IN are compact. We observe that A = τW −lim An. Let be now x = (ξ1, ξ2, ..., ξn, ...) ∈ `1 arbitrarily ﬁxed; we evaluate the distance between x and An: 5 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 267

Let be n ∈ IN suﬃciently large such that |ξn| ≤ 1. Then

d(x, An) = inf{kx − tenk; t ∈ [0, 1]} =    ∞  X  = inf |ξk| + |ξn − t|; t ∈ [0, 1] =  k=1  k6=n ( ∞ ) X = inf |ξk| − |ξn| + |ξn − t|; t ∈ [0, 1] = k=1

= kxk − |ξn| + inf{|ξn − t|; t ∈ [0, 1]} =

= kxk − |ξn| + max{−ξn, 0}.

So, it follows that d(x, An) → kxk = d(x, A). Now, if we consider sup{d(x, An) − d(x, A); x ∈ B(0, 1)} ≥ |d(en,An) − d(en,A)| = 1, so A 6= AW − lim An. Hence, A 6= τH − lim An. Still the problem has a positive answer for the monotone sequences from K(X).

Theorem 3.1. Let (X, d) be a metric space.

(i) If (An)n∈IN∗ ⊂ Cl(X) with A1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · · and A ∈ K(X) such that A = τW − lim An, then An ⊂ A ( so An are compact) ! [ A = cl An and A = τH − lim An. n∈IN

(ii) If (An)n∈IN∗ ⊂ K(X) with A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · and A ∈ Cl(X) such that A = τW − lim An, then A ⊂ An, A are compact, \ A = An and A = τH − lim An. n∈IN

∗ Proof. i) Let be k ∈ IN arbitrarily ﬁxed and u ∈ Ak. Then u ∈ An+k, ∗ ∀n ∈ IN, so d(u, An+k) = 0, ∀n ∈ IN . As A = τW − lim An it follows that d(u, A) = lim d(u, An+k), so d(u, A) = 0. From the closure property of A , n→∞ ∗ we ﬁnd that u ∈ A, i.e. Ak ⊂ A, ∀k ∈ IN , and Ak are compact. Now: e(An,A) = sup{d(x, A); x ∈ An} = 0. 268 G. APREUTESEI 6

From An ⊂ An+1 we have that d(a, An) ≥ d(a, An+1), ∀a ∈ A and then ∗ e(A, An) ≥ e(A, An+1). We denote by αn = e(A, An), n ∈ IN and by α ≥ 0 the limit of the decreasing sequence pozitive numbers (αn)n∈IN∗ . ∗ The function d(·,An) is continuous, ∀n ∈ IN and A is compact. It follows that ∃ an ∈ An such that αn = sup{d(a, An); a ∈ A} = d(an,An). ∗ ∗ From the compactness of A we ﬁnd a subsequence (ank )k∈IN ⊂ (an)n∈IN which is convergent to an element a0 ∈ A. We have:

0 ≤ d(ank ,Ank ) ≤ d(ank , a0) + d(a0,Ank ).

But d(ank , a0)−→0, d(a0,Ank )−→d(a0,A) from the τW –convergence of the ∗ sequence (An)n∈IN la A, and so αnk = d(ank ,Ank )−→0, that is α = 0. As e(An,A) = 0, we obtain that A = τH − lim An.

ii) If u ∈ X, from An ⊃ An+1 we ﬁnd that d(u, An) ≤ d(u, An+1). Partic- ularly, the sequence (d(a, An))n∈IN∗ of positive numbers is increasing ∀a ∈ A. From the τW –convergence of An to A it follows that d(a, An)−→d(a, A) = ∗ 0, and then d(a, An) = 0, i.e. a ∈ An, ∀n ∈ IN , though A ⊂ An (and A is compact). e(A, An) = sup{d(a, An); a ∈ A} = 0.

We denote by βn = e(An,A). From the decreasing monotony of the sequence (An)n∈IN∗ ; βn has the same propery. We consider β ≥ 0 its limit. The continuous funtion d(·,A) is realling its supremum on every compact An; let this supremum be xn ∈ An. Then βn = d(xn,A). Since xn ∈ An ⊂ A1 and A1 is compact, the sequence (xn)n∈IN∗ contains a subse- ∗ quence (xnk )k∈IN which is convergent to an element x0. ∗ ∀p ∈ IN, ∀k ∈ IN we have xnk+p ∈ Ank+p ⊂ Ank and since Ank is a closed set,it follows:

∗ x0 = lim xn ∈ An , ∀k ∈ IN . p→∞ k+p k

Now we will use the fact that A = τW − lim Ank and then

0 = lim d(x0,An ) = (x0,A). k→∞ k

From the continuity of the function d(·,A) and from xnk −→x0, it follows that

βnk = d(xnk ,A)−→d(x0,A) = 0, that is βn−→0. 7 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 269

The convergences e(An,A)−→0 and e(A, An)−→0 imply that A = τH − lim An.

Corollary. Let X be a metric space.On the class of the monotone sequences of substes of K(X), the following convergences coincide: `f, H, V , P , bP , AW , W and K.

Proof. For increasing monotone sequences (An)n∈IN from Cl(X) we know that exists ! [ K − lim An = cl An , n∈IN and for decreasing monotone sequences, \ K − lim An = An n∈IN ([13], p.204). Using the Theorem 3.1, we obtain that W ≡K on the monotone sequences of K(X); as H and W agree on the same class, using the implications from the Sections 1 and 2, it follows all the desired coincidences.

4. The convergence in a Mosco sense. Further we will study the coincidence between Mosco and Hausdorﬀ convergences on the classes A ⊂ Cl(X). It is necessary to presume from now that X is a linear normed space. We ﬁrst give the following

Remark. Let X be a linear normed space and S(X) the class of the singeltons of X. Then, on S(X), the convergences M, H, V, P, bP, AW, W, L reduce of the norm-convergence from X.

Lemma We consider a normed linear space X and A⊂Cl(X) a class of nonempty parts of X which is stable with respect to closed subsets, such that M =H on A. Then A⊂B(X).

Proof. From the Remark 4.1, there exists nonempty classes A on which M ≡ H. Lets suppose now that ∃ A ∈ A unbounded. Then A contains elements of a norm indeﬁnitely large:

(4.1) ∀n ∈ IN ∃ an ∈ A with kank ≥ n. 270 G. APREUTESEI 8

We consider A0 = {a1, a2, ..., an, ...}; from (4.1) it follows that the set of accumulation points for A0 is empty, so A0 is closed. As A0 ⊂ A and A is stable with respsct to closed subsets, it follows that A0 ∈ A. Similarly, the ﬁnite sets An = {a1, ..., an} ⊂ A are from A. We show that A0 = M − lim An :

1 ∀x ∈ A0 we take p ∈ IN such that x = ap. Let be the sequence ( ap, if n ≥ p, xn = , where bn ∈ An is an arbitrary element; bn, if n < p ap ∈ Ap ⊂ An for ∀n ≥ p, so xn ∈ An and, obviously, xn−→ap = x. w 2 Let be nk % ∞, xk ∈ Ank with xk −→ x.

Then the sequence (xk)k∈IN is w–bounded, so it is also bounded in norm (the strong boundedness and weak one coincide on a normed linear space: see, for ex., [?, Prop.6.11, p.147]. As {xk} ⊂ A0 and the unique bounded subsets of A0 are the singeltons, it follows that the set of the terms of the sequence {xk} is ﬁnite, so w– bounded. Therefore, x can not be only one of the terms of the sequence xk, that is x ∈ A0. So, A0 = M − lim An. Then, from the hypothesis, A0 = H −lim An, a contradiction (the Haus- dorﬀ convergence preservs the boundedness: [1], Lemma 3.1,ii).

Now we establish the moste important result concerning the Mosco and Hausdorﬀ convergences; as the most adequate ”enviroment” for M is formed by the closed convex sets, we will consider the families Ac ⊂ Cl(X).

Theorem 4.1. Let X be linear normed space. (i) If Y ⊂ X is a ﬁnite-dimensionally linear subspace,then H = M on Kc(Y ). (ii) Under the supplimentary hypothesis that X is a Hilbert space, the following assertion of i) is valid: If A ⊂ C(X) is a family which is stable with respect to close subsets, having the property that every set of A admits some elements of the greatest norm, and on whom H ≡ M, then A containsonly ﬁnite- dimensionally compact sets of X. 9 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 271

Remark. The condition which claims that the sets of A should admit some elements of maximum norm isn’t very restrictive; from the Lemma 4.1, it follows that A ⊂ B(X); then, for ex., Kc(X) satisﬁes that condition, but there are also some families strictly ampler, such as the family of the balls on X, which satisﬁes this hypothesis.

Proof of the Theorem 4.1. We first notice that Kc(Y ) = Bc(Y ) = Bc(X) ∩ Y if Y is finite-dimensionally. n i) It follows from [15, Corolar 3A]: on Kc(IR ) we have τH = τM (= τF ). ii) As A ⊂ C(X), from the Lemma 4.1 it follows that A ⊂ Bc(X). We will show that ∀A ∈ A there exists Y ⊂ X finite-dimensional such that A ⊂ Bc(Y ): Lets suppose that A is a family having the above properties, on which H = M, but A contains at least one set A ∈ Bc(X) of infinit dimension. We consider the following situations:

Case I. 0 ∈/ A. From the hypothesis, A contains at least one element having a maximum norm; let it be e1 ∈ A. We take A1 = I(e1) ∩ A, where I(B) denotes the linear envolving of the set B ⊆ X. The set A1 is closed and convex, being a finite intersection of closed convex sets. Furthermore, A1 ⊂ A, so A1 ∈ A , because A ⊂ C(X) is stable with respect to closed subsets. ⊥ We consider E1 = I(e1) orthogonal complement of the linear subspace I(e1). Then A ∩ E1 6= {0} (otherwise, A ⊂ I(e1) and A would be finite- dimensionally). We choose e2 ∈ A ∩ E1, so e1 ⊥ e2. We take A2 = I(e1, e2) ∩ A closed convex set conained in A (so A2 ∈ A) ⊥ and E2 = I(e1, e2) . Inductively, we find a set {e1, e2, ..., en, ...} ⊂ A of orthogonal elements (and, consequently, of liniarly independent elements). We denote by A0 the set: A0 = cl(I(e1, e2, ..., en, ...)) ∩ A.

As e1 an element of maximum norm in A and e1 ∈ A0 ⊂ A it follows that e1 is an element of maximum norm for A0. Eventually, we could change the numbering of the elements e1, e2, ..., en, ... in order to obtain an orthogonal sequence of decreasing norm: ke1k ≥ ke2k ≥ · · · ≥ kenk ≥ · · · . Let be An = A ∩ I(e1, e2, ..., en) a bounded, close, convex set of ﬁnite dimension, (and also compact), for every n ∈ IN∗. 272 G. APREUTESEI 10

We show that A0 = M − lim An : ∗ 1) Let be arbitrary element a ∈ A0; then there exists k ∈ IN , λ1, ..., λk ∈ k ∗ X IΓ(IΓbeing the scalar ﬁeld of X) and n1, ..., nk ∈ IN such that a = λpenp . p=1

We consider n0 = max{n1, ..., nk}. It follows that a ∈ I(e1, ..., en0 ) and, as a ∈ A, we have a ∈ An0 . We deﬁne the sequence ( a, if n ≥ n0 an = with bn ∈ An arbitrary, ∀n < n0. bn, if n < n0

Then, an ∈ An and an−→a. w 2) We ﬁx nk % ∞, ak ∈ Ank with the property ak −→ a. Since Ank ⊂ A0, and A0 is a closed convex set, so w–closed (see, for ex., [14, Prop.67, p.144]), it follows that a ∈ A0. Then, according to the hypothesis, A0 = H − lim An. We will show that A0 ∈/ H+ − lim An : ∗ For any n ∈ IN , let be an ∈ An the element of the best approximation of en+1 with respect to An : ken+1 − ank = inf{ken+1 − ak; a ∈ An} (this element exists because the set An is closed and of ﬁnite dimension). We show that the sequence (ken+1 − ank)n∈IN∗ is decreasing: Let be n−1 X an−1 = λiei ∈ An−1; since en, en+1 ⊥ an−1 we have i=1 2 2 2 ken − an−1k = kenk + kan−1k ≥ 2 2 2 2 ≥ ken+1k + kan−1k = ken+1 − an−1k ≥ ken+1 − ank ,

the last inequality is obtained from an−1 ∈ An−1 ⊂ An. We denote by `n = ken+1 − ank and by ` = lim `n ≥ 0. We prove that n→∞ n X ` > 0 : if an = µiei we have i=1 n 2 2 X 2 2 2 `n = ken+1k + |µi| keik ≥ ken+1k i=1 and so

∗ `n ≥ inf{kenk; n∈IN } ≥ inf{kak; a∈A} = d(0,A) > 0 11 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 273 because A is closed and 0∈/A, hence ` ≥ d(0,A) > 0. Now: 0 < ` ≤ ken+1 − ank = d(en+1,An) ≤

≤ sup{d(a, An); a∈A0} = e(A0,An); in conclusion, e(A0,An) 6−→ 0, so that A0 ∈/ H+ − lim An, a contradiction.

Case II. 0 ∈ A (and so 0 ∈ A0).

As A is bounded and I(e1) is unbounded, there exists a ∈ I(e1) such that a∈ / A. Then 0 ∈/ B = A − a. Let be

Bn = An − a =⇒ Bn = (I(e1, ..., en) ∩ A) − a = (A − a) ∩ I(e1, ..., en); analogically, for B0 = A0 − a we have

B0 = B ∩ cl(I(e1, ..., en, ...)).

From the case I, Bn is M–convergent at B0, but H(Bn,B) 6−→ 0. We know the inequality H(An,A0) ≥ H(An − a, A0 − a) and so A0 6= H − lim An, a contradiction. It results that A ∈ Kc(Y ), where Y is of ﬁnit dimension in X.

Remark 4.3. Let be X a Hilbert space; we want to ﬁnd a class A ⊂ Cl(X) on which the Kuratowski convergence of the net coincides with their H–convergence; this contain only compact sets of ﬁnite dimension : the proof is identically to that of the Theorem 4.1,ii), without the hypothesis of convexity. Indeed, if A0 is closed, ak ∈ Ank ⊂ A0 and ak → a, then a ∈ A0. By readjusting the proof of the Theorem 4.1 ,ii), we obtain the following result, independently relevant:

Proposition 4.1. If X is a linear normed space of infinit dimension, then for any closed linear subspace X0 ⊆ X of infinite dimension there exists a net of linear subspace of X, finite dimensionally denoted by XF ,F ∈ F, such that X0 = M − lim XF ,X0 ∈ H− − lim XF , but X0 6= H+ − lim XF .

Proof. Let be (eλ)λ∈Λ an algebraic base in X0. We consider the set F = = {F ⊂ Λ; F ﬁnit˘a} which is organised by the inclusion relation. Then F is a directed set because ∀F1,F2 ∈ F ∃ F3 = F1 ∪ F2 ∈ F such that F1 ⊆ F3 and F2 ⊆ F3. 274 G. APREUTESEI 12

We consider XF , the linear subspace (of finite dimension) generated by (eλ)λ∈F . We show that X0 ∈ M − lim XF : 1) We fix a0 ∈ X0; then ∃ λ1, ..., λp ∈ Λ and the scalars α1, ..., αp such p X that a0 = αkeλk . Let be F0 = {λ1, ..., λp}; then a0 ∈ XF0 ⊂ XF for any k=1 F ≥ F0. We define the net a0, if F ≥ F0 aF = with bF ∈ XF arbitrary. bF , in rest

k·k Obviously, aF ∈ XF , ∀F ∈ F and aF −→ a0 because ∀ε > 0 ∃ F0 ∈ F such that ∀F ≥ F0 we have kaF − a0k = 0 < ε. 2) Let be J a directed set for which ∃ ϕ : J−→F with the property ∀F ∈ F ∃ jF ∈ J such that j ≥ jF implies ϕ(j) ≥ F. w We also consider aj ∈ Xϕ(j) such that aj −→ a. But aj ∈ Xϕ(j) ⊂ X0 and, as X0 is closed (so weak closed – [14, p.144]) it results that a ∈ X0. Since XF ⊂ X0 ⊂ Sε(X0) for ∀ε > 0, is evidently that X0 ∈ H− − lim XF . We show that X0 ∈/ H+ − lim XF , namely 1 ∃ ε = > 0 such that ∀F ∈ F we have X 6⊂ S (X ): 2 0 1/2 F Let be F ∈ F and f 0 ∈ Λ\F ; we denote F 0 = F ∪ {f 0} with F 0 ⊃ F, 0 F 6= F. We suppose that aF is the element of the best approximation if ef 0 ∈ XF 0 with respect to XF , i.e. kef 0 − aF k ≤ kef 0 − bF k, for every bF ∈ XF .Since kef 0 − aF k= 6 0 we can take

1 xF 0 = (ef 0 − aF ) ∈ XF 0 − XF ⊂ X0 − X0 = X0. kef 0 − aF k

1 If b ∈ X is an arbitrary element, we prove that d(x 0 , b ) > : F F F F 2 1 kxF 0 − bF k = kef 0 − (aF + kef 0 − aF k · bF )k ≥ kef 0 − aF k 1 1 ≥ · kef 0 − aF k = 1 > kef 0 − aF k 2 13 SET CONVERGENCE AND THE CLASS OF COMPACT SUBSETS 275 because the element aF + kef 0 − aF k · bF ∈ XF . So, xF 0 ∈/ S1/2(XF ), that is X0 6⊂ S1/2(XF ).

Finally, from the Theorem 4.1 and the results of the table from the ﬁrst page ,we ﬁnd the following statement:

Corollary 4.1.

1. Let be X a normed space and Y ⊂ X a linear subspace of ﬁnite dimension. On Kc(Y ) the Mosco-convergence M agree with the locally ﬁnite one (`f) (and, in consequence , with V, P, AW, bP, L).

2. Conversely: we consider in addition the hypothesis that X is a Hilbert space; let A ⊂⊂ C(X) a class which is stable with respect to closed subsets, with the property that A contains elements of maxim norm. If the convergences M coincides with (`f) on A, then A contains only compact sets of ﬁnite domension.

Proof: 1) From [16, §10], if Y is of ﬁnite dimension, we have AW ≡ bP ≡ W ≡ K ≡ M on C(Y ). On K(X), so also on Kc(Y ), the next equalities hold: V ≡ `f ≡ P ≡ H. Moreover, on C(X) we have the implications H =⇒ L =⇒ M, and we obtain the coincidence of all those convergences. 2) On C(X), (`f) =⇒ H =⇒ M. From the hypothesis A it results that on A we have M ≡ H and we apply the Theorem 4.1; ﬁnally, we use that on Kc(Y ), lf and H are equivalent.

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Received: 10.V.2000 Faculty of Mathematics University ”Al.I.Cuza” 6600 - Ia¸si ROMANIA