ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII ”AL.I.CUZA” IAŞI
Tomul XLVII, s.I a, Matematică, 2001, f.2.
SET CONVERGENCE AND THE CLASS OF COMPACT
SUBSETS
BY
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
≡
bP
`f
⇓
≡
≡
H
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 };
{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;
S(a, ε)
=
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 }

=
\
i∈I
cl 

[
Ai  .
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.
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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 hyperconvergence 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.
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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 :
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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 − ten k; t ∈ [0, 1]} =
= inf


∞
X



|ξk | + |ξn − t|; t ∈ [0, 1]

 k=1
k6=n
(
= inf
∞
X
=


)
|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.
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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 ). Particularly, 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 subsequence (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 xnk+p ∈ Ank , ∀k ∈ IN∗ .
p→∞
Now we will use the fact that A = τW − lim Ank and then
0 = lim d(x0 , Ank ) = (x0 , A).
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.
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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 kan k ≥ n.
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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
(
xn =
ap , if n ≥ p,
, 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 Hausdorff 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 finitedimensionally compact sets of X.
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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.
i) It follows from [15, Corolar 3A]: on Kc (IRn ) 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 finitedimensionally). 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: ke1 k ≥ ke2 k ≥ · · · ≥ ken k ≥ · · · . Let be
An = A ∩ I(e1 , e2 , ..., en ) a bounded, close, convex set of finite dimension,
(and also compact), for every n ∈ IN∗ .
272
10
G. APREUTESEI
We show that A0 = M − lim An :
1) Let be arbitrary element a ∈ A0 ; then there exists k ∈ IN∗ , λ1 , ..., λk ∈
IΓ (IΓ being the scalar field of X) and n1 , ..., nk ∈ IN∗ such that a =
k
X
λp enp .
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
(
an =
a, if n ≥ n0
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 − an k = 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 − an k)n∈IN∗ is decreasing: Let be
an−1 =
n−1
X
λi ei ∈ An−1 ; since en , en+1 ⊥ an−1 we have
i=1
ken − an−1 k2 = ken k2 + kan−1 k2 ≥
≥ ken+1 k2 + kan−1 k2 = ken+1 − an−1 k2 ≥ ken+1 − an k2 ,
the last inequality is obtained from an−1 ∈ An−1 ⊂ An .
We denote by `n = ken+1 − an k and by ` = lim `n ≥ 0. We prove that
` > 0 : if an =
n
X
n→∞
µi ei we have
i=1
`2n = ken+1 k2 +
n
X
|µi |2 kei k2 ≥ ken+1 k2
i=1
and so
`n ≥ inf{ken k; n ∈ IN∗ } ≥ inf{kak; a ∈ A} = d(0, A) > 0
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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 − an k = 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 finită} 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
12
G. APREUTESEI
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
that a0 =
p
X
αk eλk . Let be F0 = {λ1 , ..., λp }; then a0 ∈ XF0 ⊂ XF for any
k=1
F ≥ F0 .
We define the net
aF =
a0 , if F ≥ F0
bF , in rest
with bF ∈ XF arbitrary.
k·k
Obviously, aF ∈ XF , ∀F ∈ F and aF −→ a0 because ∀ε > 0 ∃ F0 ∈ F such
that ∀F ≥ F0 we have kaF − a0 k = 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
∃ε=
1
> 0 such that ∀F ∈ F we have X0 6⊂ S1/2 (XF ) :
2
Let be F ∈ F and f 0 ∈ Λ\F ; we denote F 0 = F ∪ {f 0 } with F 0 ⊃ F,
F 0 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
xF 0 =
1
(ef 0 − aF ) ∈ XF 0 − XF ⊂ X0 − X0 = X0 .
kef 0 − aF k
If bF ∈ XF is an arbitrary element, we prove that d(xF 0 , bF ) >
1
:
2
1
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
kxF 0 − bF k =
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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 - Iaşi
ROMANIA