The Schur ℓ₁ Theorem for Filters
We study the classes of filters F on N such that the weak and strong F-convergence of sequences in ℓ₁ coincide. We study also an analogue of ℓ₁ weak sequential completeness theorem for lter convergence.
Gespeichert in:
| Datum: | 2007 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
|
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/7614 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | The Schur ℓ₁ Theorem for Filters / A. Aviles Lopez, B. Cascales Salinas, V. Kadets, A. Leonov // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 383-398. — Бібліогр.: 11 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-7614 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-76142016-09-28T19:16:31Z The Schur ℓ₁ Theorem for Filters Lopez, A. Aviles Salinas, B. Cascales Kadets, V. Leonov, A. We study the classes of filters F on N such that the weak and strong F-convergence of sequences in ℓ₁ coincide. We study also an analogue of ℓ₁ weak sequential completeness theorem for lter convergence. 2007 Article The Schur ℓ₁ Theorem for Filters / A. Aviles Lopez, B. Cascales Salinas, V. Kadets, A. Leonov // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 383-398. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/7614 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We study the classes of filters F on N such that the weak and strong F-convergence of sequences in ℓ₁ coincide. We study also an analogue of ℓ₁ weak sequential completeness theorem for lter convergence. |
| format |
Article |
| author |
Lopez, A. Aviles Salinas, B. Cascales Kadets, V. Leonov, A. |
| spellingShingle |
Lopez, A. Aviles Salinas, B. Cascales Kadets, V. Leonov, A. The Schur ℓ₁ Theorem for Filters |
| author_facet |
Lopez, A. Aviles Salinas, B. Cascales Kadets, V. Leonov, A. |
| author_sort |
Lopez, A. Aviles |
| title |
The Schur ℓ₁ Theorem for Filters |
| title_short |
The Schur ℓ₁ Theorem for Filters |
| title_full |
The Schur ℓ₁ Theorem for Filters |
| title_fullStr |
The Schur ℓ₁ Theorem for Filters |
| title_full_unstemmed |
The Schur ℓ₁ Theorem for Filters |
| title_sort |
schur ℓ₁ theorem for filters |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/7614 |
| citation_txt |
The Schur ℓ₁ Theorem for Filters / A. Aviles Lopez, B. Cascales Salinas, V. Kadets, A. Leonov // Журн. мат. физики, анализа, геометрии. — 2007. — Т. 3, № 4. — С. 383-398. — Бібліогр.: 11 назв. — англ. |
| work_keys_str_mv |
AT lopezaaviles theschurl1theoremforfilters AT salinasbcascales theschurl1theoremforfilters AT kadetsv theschurl1theoremforfilters AT leonova theschurl1theoremforfilters AT lopezaaviles schurl1theoremforfilters AT salinasbcascales schurl1theoremforfilters AT kadetsv schurl1theoremforfilters AT leonova schurl1theoremforfilters |
| first_indexed |
2025-07-02T10:25:44Z |
| last_indexed |
2025-07-02T10:25:44Z |
| _version_ |
1836530474993844224 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 4, pp. 383�398
The Schur `1 Theorem for Filters
Antonio Aviles Lopez
Equipe de Logique Math�ematique UFR de Math�ematiques (case 7012)
Universit�e Denis-Diderot Paris
72 place Jussieu, 75251 Paris Cedex 05, France
E-mail:avileslo@um.es
Bernardo Cascales Salinas
Departamento de Matemñticas, Universidad de Murcia
30100 Espinardo, Murcia, Spain
E-mail:beca@um.es
Vladimir Kadets and Alexander Leonov
Department of Mathematics and Mechanics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:vova1kadets@yahoo.com
leonov_family@mail.ru
Received January 16, 2007
We study the classes of �lters F on N such that the weak and strong
F-convergence of sequences in `1 coincide. We study also an analogue of `1
weak sequential completeness theorem for �lter convergence.
Key words: Schur `1 theorem, weak sequential completeness, �lter con-
vergence, �lter, statistical convergence.
Mathematics Subject Classi�cation 2000: 46B25, 54A20.
1. Preliminaries
Every theorem of Classical Analysis, Functional Analysis or of the Measure
Theory that states a property of sequences leads to a class of �lters for which
this theorem is valid. Sometimes this class of �lters is trivial (say, all �lters
or the �lters with countable base), but in some cases this approach leads to a
new class of �lters, and the characterization of this class can be a rather non-
trivial task. Among these non-trivial examples there are Lebesgue �lters (for
which the Lebesgue dominated convergence theorem is valid), Egorov �lters which
The work of the third Author was supported by the Seneca Foundation, Murcia.
Grant No. 02122/IV2/05.
c
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov, 2007
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
correspond to the Egorov theorem on almost uniform convergence [7], and those
�lters F for which every weakly F convergent sequence has a norm-bounded
subsequence [6].
One of the reasons to study these problems is that they throw a new light
on classical results. Say, it is known that the dominated convergence theorem
can be deduced from the Egorov theorem. The question, whether the converse
is true, has no sense in the classical context: if both the theorems are true, how
can we see that one of them is not deducible from the other one? But, if one
looks at the correspondent classes of �lters, the problem makes sense and in fact
there are Lebesgue �lters which are not Egorov ones (in particular the statistical
convergence �lter).
In this paper we study the Schur theorem on the coincidence of weak and
strong convergence in `1 in a general setting when the ordinary convergence of
sequences is substituted by a �lter convergence. We show that for some �lters this
theorem is valid and for some is not and give necessary and su�cient conditions
(close one to another) for its validity. After that we consider the Schur theorem
for ultra�lters. We also study a related problem of weak sequential completeness
for �lter convergence.
Recall that a �lter F on a set N is a nonempty collection of subsets of N
satisfying the following axioms: ; =2 F ; if A;B 2 F , then A
T
B 2 F ; and for
every A 2 F if B � A, then B 2 F . All over the paper if the contrary is not
stated directly we consider �lters on a countable set N . Sometimes for simplicity
we put N = N.
A sequence (xn), n 2 N in a topological space X is said to be F-convergent
to x (and we write x = F � limxn or xn !F x) if for every neighborhood U of x
the set fn 2 N : xn 2 Ug belongs to F .
In particular, if one takes as F the �lter of those sets whose complement is
�nite (the Fr�echet �lter), then F-convergence coincides with the ordinary one.
The natural ordering on the set of �lters on N is de�ned as follows: F1 � F2
if F1 � F2. If G is a centered collection of subsets (i.e., all �nite intersections of
the elements of G are nonempty), then there is a �lter containing all the elements
of G. The smallest �lter containing all the elements of G is called a �lter generated
by G.
Let F be a �lter. A collection of subsets G � F is called the base of F if for
every A 2 F there is a B 2 G such that B � A.
A �lter F on N is said to be free if it dominates the Fr�echet �lter. All the
�lters below are supposed to be free. Thus every ordinary convergent sequence
will be automatically F-convergent.
A maximal in the natural ordering �lter is called an ultra�lter. The Zorn
lemma implies that every �lter is dominated by an ultra�lter. A �lter F on N is
an ultra�lter if and only if for every A � N either A or N n A belongs to F .
384 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
A subset of N is called stationary with respect to a �lter F (or just F-statio-
nary) if it has nonempty intersection with each member of the �lter. Denote the
collection of all F-stationary sets by F�. For an I 2 F� we call the collection of
sets fA\ I : A 2 Fg the trace of F on I (which is evidently a �lter on I), and by
F(I) we denote the �lter on N generated by the trace of F on I. Clearly F(I)
dominates F . Any subset of N is either a member of F or the complement of a
member of F , or the set and its complement are both F -stationary sets. F� is
precisely the union of all ultra�lters dominating F . F� is a �lter base if and only
if it is equal to F and F is an ultra�lter.
Theorem 1.1. Let X be topological space, xn, x 2 X and let F be a �lter
on N . Then the following conditions are equivalent:
(1) (xn) is F-convergent to x;
(2) (xn) is F(I)-convergent to x for every I 2 F�;
(3) x is a cluster point of (xn)n2I for every I 2 F�.
P r o o f. Implications (1) ) (2) and (2) ) (3) are evident. Let us prove
that (3) ) (1) do not F-converge to x. Then there is such a neighborhood U
of x that in each A 2 F there is a j 2 A
`
such that xj 62 U . Consequently
I = fj 2 N : xj 62 Ug is stationary and x is not a cluster point of (xn)n2I .
More about �lters, ultra�lters and their applications one can �nd in most of
advanced General Topology textbooks, for example, in [10].
For the standard Banach space terminology we refer to [8]. All the spaces,
functionals and operators (although this does not matter) are assumed to be over
the �eld of reals. Before we pass to the main results let us recall some notations
and geometric properties of `1. Denote by en the n-th element of the canonical
basis of `1 and by e�n the n-th coordinate functional on `1. In this notations for
every x 2 `1 we have
x =
X
n2N
e�n(x)en:
Recall that en are separated from 0 by the functional f(x) =
P
n2N e
�
n(x), i.e., 0
is not a weak cluster point of (en). The following lemma can be easily extracted
from the block-basis selection method (see [8], volume 1). We give the proof for
completeness.
Lemma 1.2. Let yn 2 `1, infn2N kynk = "0 > " > 0 and let fm(n)g be an
increasing sequence of naturals. Denote zi =
P
k2(m(i);m(i+1)] e
�
k
(yi)ek. If under
these notations supn2N kyn � znk < "=2 (i.e., (yn) is a small perturbation of the
block-basis (zn)), then (yn) is equivalent to the sequence (kynken) and consequently
0 is not a weak cluster point of (yn).
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 385
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
P r o o f. We must �nd c1; c2 > 0 such that for every collection of scalars an
c1
X
n2N
janjkynk �
X
n2N
anyn
� c2
X
n2N
janjkynk:
The upper estimate with c2 = 1 follows immediately from the triangular inequa-
lity. The lower one holds with c1 = 1� "0="
X
n2N
anyn
�
X
n2N
janjkznk �
X
n2N
janjkyn � znk
�
X
n2N
janjkynk � 2
X
n2N
janjkyn � znk �
�
1� "
"0
�X
n2N
janjkynk:
2. Simpli�ed Schur Property for Filters
There are several natural ways to generalize the Schur theorem for �lters
instead of sequences. Let us start with the one leading to a class of �lters which
we are able to characterize completely in combinatorial terms.
De�nition 2.1. A �lter F on N is said to be a simple Schur �lter (or is said
to have the simpli�ed Schur property) if for every coordinate-wise convergent to 0
the sequence (xn) � `1 if (xn) weakly F-converges to 0, then F � lim kxnk = 0.
For an in�nite set I � N let us call a blocking of I a disjoint partition D =
fDkgk2N of I into nonempty �nite subsets.
De�nition 2.2. A �lter F on N is said to be block-respecting if for every
I 2 F� and for every blocking D of I there is a J 2 F�, J � I such that
jJ \ Dkj = 1 for all k, where the �modulus� of a set stands for the number of
elements in the set.
Remark 2.3.
8k2NjJ \Dkj � 1 (2.1)
instead of jJ \Dkj = 1, one will obtain an equivalent de�nition.
Remark 2.4. If F is block-respecting, then F(J) for every J 2 F� is also
block-respecting.
Lemma 2.5. Let F be a block-respecting �lter and let (xn) � `1 form
a coordinate-wise convergent to 0 sequence, which does not F-converge to 0 in
norm. Then there is a J 2 F�; such that the sequence (xn); n 2 J is equivalent to
(aiei), where ei form the canonical basis of `1, ai � 1.
386 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
P r o o f. Due to Th. 1.1 there is an I 2 F� such that infn2I kxnk > " > 0.
Fix a decreasing sequence of Æk > 0,
P
k2N Æk � "=8. Using the de�nition of `1 let
us select an increasing sequence of naturals (m(n)) and such that for every n 2 NX
k�m(n)
je�k(xn)j < Æn; (2.2)
and using the coordinate-wise convergence of xn to 0 select an increasing sequence
of integers (ni) such that n0 = 0, Di := (ni�1; ni]\ I 6= ; and for every i 2 N and
j � ni+1 X
k�m(ni)
je�k(xj)j < Æi: (2.3)
Taking in account the respect which F has to the blocks Di let us select a J =
fj1; j2; : : :g 2 F�; J � I such that ji 2 (ni�1; ni] for all i 2 N. Since J 2 F�; either
J1 = fj1; j3; j5 : : :g or J2 = fj2; j4; j6 : : :g is an F-stationary set as well. Let, say,
J2 2 F�. Let us show that in fact vectors yi = xj2i are small perturbations of the
block-basis zi =
P
k2(m(n2(i�1));m(n2i)]
e�
k
(yi)ek, which due to Lem. 1.2 completes
the proof. So,
kyi � zik =
X
k�m(n2i�2)
je�k(xj2i)j+
X
k>m(n2i)
je�k(xj2i)j:
Taking into account inequalities (2.2), (2.3) and that j2i 2 (n2i�1; n2i], we get
kyi � zik � 2Æj2i which implies the condition of Lem. 1.2.
Theorem 2.6. A �lter F on N has the simpli�ed Schur property if and only
if F is block-respecting.
P r o o f. The �if� part of the theorem follows immediately from Lem. 2.5.
So let us turn to the �only if� part. Assume that F is not block-respecting, i.e.,
there is an I 2 F� and there is a blocking D of I such that every J � I satisfying
(2.1) is not F-stationary. In other words N n J 2 F for every J � I satisfying
(2.1). Since the �nite intersection of the �lter elements again belongs to F , we
can reformulate the fact that F is not block-respecting as follows: there is an
I 2 F� and such a blocking D = fDkgk2N of I that N n J 2 F for every J � I
satisfying
sup
k2N
jJ \Dkj <1: (2.4)
Now, using Dvoretzky's almost Euclidean section theorem, let us select an in-
creasing sequence of integers 0 = m0 < m1 < m2 < : : : and a sequence of vectors
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 387
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
xn 2 `1 such that xn = 0 when n 62 I; xn 2 Linfekgk2(mi�1;mi] when n 2 Di, and
for every collection of scalars an0
@X
n2Di
janj2
1
A
1=2
�
X
n2Di
anxn
�
0
@2
X
n2Di
janj2
1
A
1=2
: (2.5)
This sequence converges coordinate-wisely to 0 and is not F -convergent to 0 in
norm, because kxnk � 1 for every n 2 I. Let us prove xn's weak F-convergence
to 0, which will show that F does not have the simpli�ed Schur property. Well,
take an f 2 `�1 with kfk = 1, �x an " > 0 and consider the set of indexes
A = fn : f(xn) < "g. We must prove that A 2 F . Since the complement of
A lies in I, it is su�cient to show that J = N n A = fn : f(xn) � "g satis�es
(2.4). In other words, we must estimate dk = jJ \Dkj from above uniformly in k.
Let us do this. Consider yk =
P
n2J\Dk
xn. Then f(yk) � "dk and due to (2.5)
kykk2 � 2dk. Hence
"dk � f(yk) �
p
2dk
and dk � 2="2.
Remark 2.7. One can see that in the �only if" part of the Th. 2.6 proof the
sequence (xn) is bounded by
p
2. So, if one restricts the Def. 2.1 to the bounded
sequences, the class of �lters does not change. In fact this is a little bit surprising
because a weakly F convergent sequence can converge to in�nity in norm [6]. If one
analyzes the characterization [6] of those �good" �lters F for which every weakly
F convergent sequence has a norm-bounded subsequence, one can see that every
simple Schur �lter is �good". The only obstacle to see this without refereeing to
[6] is the coordinate-wise convergence which appears in Def. 2.1. To see that this
obstacle is not fatal one really needs to go to the proofs of [6].
3. Schur Filters
Let us pass now to the study of the most natural Schur theorem generalization,
which is easier to formulate, but is much more complicated to characterize in
combinatorial terms.
De�nition 3.1. A �lter F on N is said to be a Schur �lter (or is said to
have a Schur property) if for every weakly F-convergent to 0 sequence (xn) � `1,
n 2 N the F � limkxnk equals 0.
Evidently, every Schur �lter has the simpli�ed Schur property. By now we do
not know if the converse holds true as well.
To simplify the exposition we mostly consider N = N, but the general case
cannot di�er from this particular one.
388 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
De�nition 3.2. F is said to be a diagonal �lter if for every decreasing sequence
(An) � F of the �lter elements and for every I 2 F� there is a J 2 F�, J � I
such that jJ n Anj <1 for all n 2 N.
Lemma 3.3. If a �lter F on N is diagonal, then for every I 2 F� and for
every coordinate-wise F-convergent to 0 sequence (xn) � `1 there is a J 2 F�,
J � I such that xn coordinate-wisely converge to 0 along J .
P r o o f. Fix a decreasing sequence of subsets Un, forming a base of neigh-
borhoods of 0 in the topology of coordinate-wise convergence. De�ne An = fk 2
N : xk 2 Ung. Since F is diagonal there is a J 2 F�, J � I such that jJ nAnj <1
for all n 2 N. This is the J we desire.
Remark 3.4. As one can see from the proof, the only property of the coordi-
nate-wise convergence topology we used is the countable base of 0 neighborhoods
existence. Also one can easily prove the inverse to the Lem. 3.3 result: if F is not
diagonal, then there is a I 2 F� and a coordinate-wise F-convergent to 0 sequence
(xn) � `1 such that for every J 2 F�, J � I the sequence (xn) does not converge
coordinate-wisely to 0 along J .
Let us demonstrate this inverse theorem. By the negation of the diagonality
de�nition a decreasing sequence of An 2 F and an I 2 F� exist such that if S � I
satis�es the condition jS n Anj < 1 for all n 2 N, then N n S 2 F . Without
loss of generality, one may assume that all the Dn := An+1 n An are in�nite andS
n
Dn = I. Then every J 2 F�, J � I must satisfy the condition
jfn 2 N : jJ \Dnj =1gj =1: (3.1)
For every n 2 I denote by f(n) such index that n 2 Df(n). Consider the following
sequence (xn): for n 2 N n I put xn = 0, and for n 2 I put xn = en + ef(n). This
sequence is the one we need.
Theorem 3.5. If a �lter F on N is diagonal and block-respecting, then F has
the Schur property.
P r o o f. Let (xn) � `1 be weakly F-convergent to 0. Arguing �ad absurdum�
suppose that there is an I 2 F� such that
inf
n2I
kxnk > " > 0: (3.2)
Due to Lem. 3.3 there is a J 2 F�, J � I such that xn coordinate-wisely converge
to 0 along J . Since F(J) is block-respecting (Remark 2.4), the condition (3.2)
contradicts Th. 2.6.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 389
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
It was shown in Th. 2.6 that the block-respect of F was a necessary condition in
order to be a Schur �lter. Our next goal is to show that the diagonality of F is not
a necessary condition. To do this de�ne a special �lter on N . LetD = fDngn2N be
a disjoint partition of N into in�nite subsets. For every sequence C = fCngn2N of
�nite subsets Cn � Dn and everym 2 N introduce the setBm;C =
S1
n=m(DnnCn).
The sets Bm;C form a �lter base. Denote the corresponding �lter by FD. One
can easily see that FD is an example of a non- diagonal block-respecting �lter. In
fact this �lter �almost" appeared in Remark 3.4. To make the picture clearer, we
may represent N as an in�nite matrix N � N, with Dn = f(k; n) : k 2 Ng being
its columns.
De�nition 3.6. A �lter F on N is said to be self-reproducing if for every
I 2 F� there is a J 2 F�, J � I such that the structure of the trace of F on J is
the same as of the original �lter F , i.e., there is a bijection s : N ! J that maps
F onto its trace on J : A 2 F () s(A) 2 F(J).
Theorem 3.7. FD is a Schur �lter, i.e., diagonality is not a necessary con-
dition for the �lter's Schur property.
P r o o f. First, remark that a subset J � N is FD-stationary if and only
if condition (3.1) is met. In particular, for every in�nite subset M � N and for
every selection of in�nite subsets An � Dn, n 2 M the set
S
n2M An is an FD-
stationary set. Let us call such sets of the form
S
n2M An �standard sets�. Every
FD-stationary set contains a standard subset. Remark also that the structure of
the trace of FD on a standard subset J is exactly the same as of the original �lter
FD, i.e., FD is self-reproducing.
To prove the theorem assume contrary that there is a sequence (xn) � `1,
n 2 N that FD-weakly converges to zero but the norms do not FD-converge to
zero. So there is an " > 0 and such a standard set J � N , that kxnk � "
for all n 2 J . According to the previous remark, we may assume without loss
of generality that J = N , i.e kxnk � " for all n 2 N . Passing from xn to
xn=kxnk, we may suppose that kxnk = 1 for all n. For every �xed m 2 N select a
subsequence of D0
m � Dm such that xn; n 2 D0
m coordinate-wisely converge to an
element ym 2 `1. Passing to a new standard set of indexes
S
m2N D
0
m, we reduce
the situation to the case when xn; n 2 Dm converge coordinate-wisely to ym for
every m 2 N.
Notice that due to the weak FD-convergence to zero of the whole sequence
(xn); n 2 N , ym converge coordinate-wisely to zero. In fact, for arbitrary coordi-
nate functional e�
k
and for every " > 0 there is a set of the form Bm;C such that
je�
k
(xj)j < " for all j 2 Bm;C . This means that for i 2 N, i > m we have
je�k(yi)j = lim
j2Di
je�k(xj)j � ":
390 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
This means in its turn the desired coordinate-wise convergence to zero of (ym).
Introduce some notation: for n 2 N denote by f(n) such index that n 2 Df(n).
Put zn = xn� yf(n). Consider two cases. The �rst one: kznk !FD 0. In this case
kymk ! 1 asm!1, but on the other hand the condition yf(n) = xn�zn w!FD 0
implies ordinary weak convergence of (ym) to 0, which is impossible according to
the Schur theorem. In the remaining case, there is a standard set on which kznk
are bounded from below, so we may again without loss of generality assume that
kznk > " > 0 for all n 2 N .
Claim. There is such a standard set J � N that the sequence (zn)n2J is
equivalent to the canonical basis of `1.
P r o o f o f t h e c l a i m. Fix a decreasing sequence of Æk > 0, k 2 N ,P
k2N Æk � "=8. Using the de�nition of `1 let us select naturals m(n) such that
for every n 2 N the condition X
k�m(n)
je�
k
(zn)j < Æn
holds true. Take an arbitrary n1 2 D1. Now using consequently the coordinate-
wise convergence to 0 of sequences (zn), n 2 Dm for values of m = 1; 2, 1; 2; 3,
1; 2; 3; 4; : : : select a sequence (ni) � N in such a way that n2 2 D1, n3 2 D2,
n4 2 D1, n5 2 D2, n6 2 D3, etc. (like triangle enumeration of a matrix) and for
every i 2 N X
k�s(i)
je�k(zni+1
)j < "
2i+3
;
where s(i) denotes maxk�im(nk). Under this construction J = (ni)i2N is a stan-
dard set, and zni is just a small perturbation of the block-basis
wi =
X
k2(s(i�1);m(ni)]
e�k(zni)ek;
which due to Lem. 1.2 means that the claim is proved.
Now the last step. Once more without loss of generality assume that J � N
from the Claim in fact equals N , i.e., (zn)n2N is equivalent to the canonical basis
of `1. Then for every bounded sequence of scalars (an)n2N there is a functional
x� 2 `�1 such that x�(zn) = an for all n 2 N . Select these an = �1 in such a way
that for every i 2 N
jfn 2 Di : an = 1gj = jfn 2 Di : an = �1gj =1:
Then for the corresponding functional x� we have for every i 2 N
lim sup
n2Di
x�(xn)� lim inf
n2Di
x�(xn) = lim sup
n2Di
x�(zn)� lim inf
n2Di
x�(zn) = 2;
which contradicts the weak FD-convergence of xn.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 391
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
4. Category Respecting and Strongly Diagonal Filters
and Ultra�lters
Let us introduce one more class of �lters, which are block-respecting and
diagonal at the same time.
De�nition 4.1. F is said to be strongly diagonal if for every decreasing se-
quence (An) � F of the �lter elements and for every I 2 F� there is a J 2 F�,
J � I such that
j(J \An) n An+1j � 1 for all n 2 N: (4.1)
According to Th. 3.5 all strongly diagonal �lters have the Schur property.
De�nition 4.2. A �lter F on N is said to be category respecting if for every
compact metric space K and for every family of closed subsets (FA)A2F of K, if
FA � FB ; whenever B � A in F ;
and K =
S
A2F FA, then int(FB) 6= ; for some B 2 F .
The obvious examples of category respecting �lters are those of countable base.
Moreover, every �lter with a base of cardinality k < m is category respecting (see
[5, p. 3�4] for the de�nition of m and Th. 13A, p. 16 for the corresponding result).
But the Martin axiom means that m equals the cardinality of continuum so, if we
accept the Martin axiom together with the negation of the continuum hypothesis,
we can go to some �lters with the uncountable base.
The proof of the Schur property for `1 by using the Baire theorem, as presented
in [3, Prop. 5.2], gives a hint that the category respecting �lters are related to the
Schur property. The next theorem shows that in fact to be category respecting is
a stronger restriction than to have the Schur property.
Theorem 4.3. If F is a category respecting �lter on N, then F is strongly
diagonal.
P r o o f. Assume contrary that F is not strongly diagonal, i.e., there is
a decreasing sequence (An) � F of the �lter elements and there is an I 2 F�
such that for all J 2 F�, J � I the condition (4.1) is not met. Without loss of
generality, we may assume that the �lter is de�ned only on I (pass to the trace
of F on I), that
T
n2N An = ; (this intersection is not stationary, so we may just
erase this intersection from I) and that all Dn := An nAn+1 are not empty. If one
picks up a sequence of �nite subsets
Cn � Dn; sup
n2N
jCnj <1; then N n
[
n2N
Cn 2 F : (4.2)
392 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
Let us introduce the following compact topological spaces eDn: if Dn is �nite,
then eDn = Dn with discrete topology; if Dn is in�nite, then eDn = Dn
Sf1ng �
the one-point compacti�cation of Dn. Recall that K =
Q
n2N
eDn is compact in
coordinate-wise convergence topology and metrizable. De�ne a family of closed
sets (FA)A2F in K as follows:
FA = fx 2 K : �n(x) 2 eDn n A for all n 2 Ng;
where �n : K ! eDn stands for the n-th coordinate projection. These sets are
closed and have empty interior (the interior can be non-empty only if for a suf-
�ciently large m Dn \ A = ; for all n � m, which is not the case becauseS
k�mDk = Am 2 F). For every x 2 K the set A(x) = N n S
n2Nf�n(x)g is
a �lter element (due to (4.2)) and x 2 FA. So the union of all (FA)A2F equals K.
Contradiction.
Corollary 4.4. If F is a category respecting �lter on N, then F is a Schur
�lter.
Corollary 4.5. Every �lter with a countable base is strongly diagonal.
Theorem 4.6. Under the assumption of continuum hypothesis there is
a strongly diagonal ultra�lter.
P r o o f. Denote by
the set of all countable ordinals. Let us enumerate
as (I(�); A(�)); � 2
all the pairs (I;A), where I is an in�nite subset of N,
and A is a decreasing sequence of in�nite subsets of N: A(�) = (An(�))n2N ,
N � A1(�) � A2(�) : : :. We construct recurrently an increasing family F�; � < !1
of �lters with countable base and an increasing family of sets
� �
as follows:
F1 is the Frech�et �lter,
1 = ;. If we have an ordinal of the form �+1, we proceed
as follows: we �nd the smallest � 2
n
� such that I(�) 2 F�� and such that
A(�) consists of F� elements. Applying Cor. 4.5, we �nd a J 2 F��; J � I = I(�)
such that (4.1) holds true for An = An(�). De�ne F�+1 as the �lter generated
by F� and J , and put
�+1 =
� [ f�g.
If � is a limiting ordinal, put F� =
S
�<�
F� and
� =
S
�<�
�.
De�ne the �lter F we need as F =
S
�<!1
F� . Let us demonstrate that F
is an ultra�lter. To do this we must prove that F� � F . Let B 2 F�. Then
B 2 F�� for all �. Let � 2
be the smallest ordinal, for which I(�) = B and
A(�) consists of the �lter F elements. Then there is an �, for which all An(�)
belong to F�. If � 2
� this means that the pair (I(�); A(�)) has appeared in
our recurrent construction, and a subset J of B (and hence B itself) was added to
the �lter. If not, then not later than on the step �+ 1 + � this pair (I(�); A(�))
has appeared in our recurrent construction and a subset J of B was added to the
�lter. By the same argument F is strongly diagonal.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 393
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
Notice that the diagonality of an ultra�lter F is equivalent to the following
well-known property: F is a �P-point of �N". The consistency of P-points non-
existence is a celebrated result of Shelah [11]. So, since every strongly diagonal
�lter is diagonal some set theoretic assumption is needed for the last theorem.
By the way in the setting of ultra�lters a property equivalent to �block-respect",
called �Q-point of �N" was also studied and the non-existence of Q-points is also
known to be consistent [9].
To conclude this section let us present an example of a strongly diagonal �lter
which is not category respecting. This example resembles strongly the proof of
Th. 4.3. Let D = fDngn2N be a disjoint partition of N into in�nite subsets.
For every sequence C = fCngn2N of �nite subsets Cn � Dn introduce the set
BC =
S
n2N(Dn nCn). The sets BC form a �lter base. Denote the corresponding
�lter by Fd. A set J � N is Fd-stationary if and only if there is an n 2 N such
that jJ \Dnj =1. One can easily see that Fd is strongly diagonal. To show that
it is not category respecting consider the same system of subsets (FA)A2F of the
same compact K as in the proof of Th. 4.3. The only di�erence is that now in the
de�nition of K we do not need to consider the case of �nite Dn. These sets FA
are closed, they have empty interior, but their union contains all the K, which is
impossible if Fd is category respecting.
5. Weak Sequential Completeness Theorem for Filters
De�nition 5.1. A �lter F on N is said to be weak `1 complete (or in abbrevi-
ated form WC1-�lter) if for every F-convergent in the topology �(`��1 ; `
�
1) bounded
sequence (xn) � `1 its weak* limit x 2 `��1 in fact belongs to `1.
It is known that every Banach space with the Schur property is weakly sequen-
tially complete. The next theorem together with Th. 4.6 shows that the picture
for �lters is more colorful.
Theorem 5.2. An ultra�lter cannot be weak `1 complete.
P r o o f. Let F be a (free as always) ultra�lter on N. Consider an arbitrary
f = (f1; f2; : : :) 2 `1 = `�1. Then for the canonical basis (en) of `1 we have
lim
F
f(en) = lim
F
fn;
which shows that the sequence (en) weakly* converges to the functional limF on
`1, which evidently does not belong to `1.
To show that aWC1-�lter may have no Schur property (and even to be without
the simpli�ed Schur property), let us recall some elements of statistical conver-
gence theory [4, 2].
394 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
A sequence (xk) in a topological space X is statistically convergent to x if for
every neighborhood U of x the set fk : xk 62 Ug has natural density 0, where the
natural density of a subset A � N is de�ned to be Æ(A) := limn n
�1jfk � n :
k 2 Agj.
Denote Fs = fI � N : Æ(N n I) = 0g and remark that Fs is a �lter. As it is
easy to see, an Fs-convergence and statistical convergence coincide, and a set J
is Fs-stationary provided Æ(J) 6= 0.
Recall that a scalar sequence (xk) is said to be strongly Cesaro-summable if
there is a scalar x such that
lim
n!1
1
n
nX
j=1
jx� xj j = 0:
It is known that a bounded scalar sequence is statistically convergent if and
only if it is strongly Cesaro-summable (for a general version of this criterion see
[1, Th. 8]). Let us apply this fact.
Theorem 5.3. Fs is a WC1-�lter but does not have the simpli�ed
Schur property.
P r o o f. Consider the blocking of N into Dn = (2
n�1; 2n+1
]. Every set J � N
intersecting each of Dn by no more than one element, has zero natural density
and consequently cannot be Fs-stationary. Hence Fs is not block-respecting and
by Th. 2.6 Fs does not have the simpli�ed Schur property.
Let us show now the weak `1 completeness of Fs. Let (xn) � `1 be a bounded
sequence and let weak* Fs-limit of (xn) be equal to an x�� 2 `��1 . This means
that for every f 2 `�1
lim
n!1
1
n
nX
j=1
jf(x�� � xj)j = 0:
Hence the vectors 1
n
P
n
j=1 xj weakly*-converge to x
�� as n!1. By the ordinary
weak sequential completeness of `1 this means that x�� 2 `1.
Our next goal is to show that if one avoids ultra�lters in a reasonable sense,
then the same su�cient condition, which we have for the Schur property, works
for the WC1 as well.
De�nition 5.4. A �lter F on N is said to be a paper �lter (p-�lter) if all
traces of F on F-stationary subsets are not ultra�lters.
Theorem 5.5. If a p-�lter F on N is diagonal and block-respecting, then F
is a WC1-�lter.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 395
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
P r o o f. Let (xn) � `1 be a bounded sequence and let F-limit of (xn) in the
topology �(`��1 ; `
�
1) be equal to an x
�� 2 `��1 n `1. Consider the standard projection
P : `��1 ! `1, which maps every element of `��1 (i.e., a linear functional on `1)
into its restriction on c0. Denote x = Px��. Without loss of generality, we may
assume that x = 0: otherwise consider xn � x instead of xn. This assumption
means that xn coordinate-wisely converge to 0 with respect to the �lter F . Due to
Lem. 3.3 there is an I 2 F�, such that xn coordinate-wisely converge to 0 along I.
Since F(I) is block-respecting (Remark 2.4), we may apply Lem. 2.5 to get such
a J 2 F� J � I, that the sequence (xn), n 2 J is equivalent to the canonical basis
of `1 (here we also use the boundedness of the sequence). Since F(J) is not an
ultra�lter we can decompose J into two disjoint F -stationary subsets J1 and J2.
Consider a functional x� 2 `�1 which takes value 1 on all xn, n 2 J1 and is equal
to �1 on every xn, n 2 J2. Then
1 = lim
F(J1)
x�(xn) = x�(x��) = lim
F(J2)
x�(xn) = �1:
This contradiction completes the proof.
To proceed further let us introduce the sum and the product of �lters.
De�nition 5.6. Let F1, F2 be �lters on N1 and N2, respectively. De�ne
F1 + F2 as the �lter on N1 [ N2 consisting of those elements A � N1 [N2 that
A \ N1 2 F1 and A \ N2 2 F2. The �lter F1 � F2 is de�ned on N1 � N2 with
base formed by the sets A1 �A2, A1 2 F1, A2 2 F2.
De�nition 5.7. A �lter F on N is said to have the double Schur property if
F � F is a Schur �lter.
Theorem 5.8. Every �lter F with the double Schur property is a WC1-�lter
and a Schur �lter at the same time.
P r o o f. Consider such a bounded sequence (xn) � `1 that F -limit of (xn)
in topology �(`��1 ; `
�
1) is equal to x
�� 2 `��1 . Then the double sequence (xn � xm)
is weakly F � F convergent to 0. According to the double Schur property of F
this implies that kxn � xmk !F�F 0, i.e., (due to the completeness of `1) there
is an element x 2 `1 such that kxn � xk !F 0. Evidently x�� = x 2 `1.
6. Domination by Schur and WC1 Filters. Open Problems
De�nition 6.1. A property P of �lters (or a corresponding class of �lters) is
said to be quasi-increasing if for every F 2 P all the �lters of the form F(J) for
every J 2 F� have the property P as well.
396 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
The Schur `1 Theorem for Filters
Remark 6.2. F(J)-convergence to 0 (in arbitrary �xed topology) of a sequence
(xn) is equivalent to F-convergence to 0 in the same topology of the sequence
(xn�J(n)). Consequently the properties de�ned only through convergence to 0
(like the Schur or double Schur properties) are quasi-increasing.
De�nition 6.3. A property P of �lters is said to be decreasing if for every
F 2 P all the �lters dominated by F have the property P as well.
Evidently WC1 �lters form a decreasing class. So one can improve Th. 5.8
as follows: every �lter dominated by a �lter with the double Schur property is a
WC1-�lter. This is an improvement, because of the following proposition:
Theorem 6.4. The Schur property, the double Schur property and moreover
every nontrivial quasi-increasing property P of �lters are not decreasing.
P r o o f. Let F1 2 P, F2 62 P be �lters on N1 and N2, respectively. Then
F = F1 + F2 is a �lter on N1 [ N2 which cannot have the property P, because
F(N2) 62 P. On the other hand F(N1) 2 P but F(N1) dominates F .
One can introduce a bit weaker but still reasonable version of the Schur pro-
perty, which is decreasing:
De�nition 6.5. A �lter F on N is said to be an almost Schur �lter (or is said
to have the almost Schur property) if for every weakly F-convergent to 0 sequence
(xn) � `1, n 2 N the norms of xn are not separated from 0 (or in other words 0
is a cluster point for kxnk, n 2 N).
Theorem 1.1 easily implies that a �lter F on N has the Schur property if and
only if all the �lters F(J) for every J 2 F� are almost Schur �lters.
One can also introduce increasing properties:
De�nition 6.6. A property P of �lters is said to be increasing if for every
F 2 P all the �lters that dominate F have the property P as well.
Evidently the negation of an increasing property is a decreasing one and contra
versa.
De�nition 6.7. Let P be an increasing (decreasing) class of �lters. A class of
�lters P1 � P is said to be a basis for P if P is the smallest increasing (decreasing)
class, containing P1.
The problem, which looks interesting, is to construct explicitly a class of al-
most Schur �lters, which forms a base for the class of all almost Schur �lters.
The same question makes sense for the negation of property to be almost Schur
�lter. An analogous study was done in [6] for the class of those �lters F for
which the weak F-convergence of a sequence implies the existence of a bounded
subsequence.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4 397
A. Aviles Lopez, B. Cascales Salinas, V. Kadets, and A. Leonov
References
[1] J. Connor, Two Valued Measures and Summability. � Analysis 10 (1990), 373�385.
[2] J. Connor, A Topological and Functional Analytic Approach to Statistical Con-
vergence. � Analysis of Divergence, (Orono, ME, 1997); Appl. Numer. Harmon.
Anal., Birkhauser, Boston, MA, 1999, 403�413.
[3] J.B. Conway, A Course in Functional Analysis. � Graduate Texts Math., 96.
Springer�Verlag, New York, 1985. MR 86h:46001.
[4] H. Fast, Sur la Convergence Statistique. � Colloq. Math. 2 (1951), 241�244.
[5] D.H. Fremlin, Consequences of Martin's Axiom. � Cambridge Tracts Math., 84.
Cambridge Univ. Press, Cambridge, 1984. MR780933 (86i:03001).
[6] M. Ganichev and V. Kadets, Filter Convergence in Banach Spaces and Generalized
Bases. General Topology in Banach Spaces (Taras Banakh, Ed.). NOVA Sci. Publ.,
Huntington, New York, 2001, 61�69.
[7] V. Kadets and A. Leonov, Dominated Convergence and Egorov Theorems for Filter
Convergence. � J. Math. Phys., Anal., Geom. 3 (2007), 196�212.
[8] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II. Springer�Verlag,
Berlin, 1996.
[9] A. W. Miller, There are no Q-points in Laver's Model for the Borel Conjecture. �
Proc. Amer. Math. Soc. 78 (1980), No. 1, 103�106.
[10] S. Todorcevic, Topics in Topology. � Lecture Notes Math. 1652. Springer�Verlag,
Berlin, 1997.
[11] E.L. Wimmers, The Shelah P -point Independence Theorem. � Israel J. Math. 43
(1982), No. 1, 28�48.
398 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 4
|