Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 625-632
625
THE CLOSED NEIGHBORHOOD AND FILTER CONDITIONS
IN SOLID SEQUENCE SPACES
P.D. JOHNSON, JR.
Department of Algebra, Comb[natorlcs, and Analysis
120 Math Annex
Auburn University
Auburn, Alabama, 36849, U.S.A.
(Received June 30, 1988)
ABSTRACT.
Let E be a topological vector space of scalar sequences, with topology
(E,T) satisfies
the
closed
neighborhood
,
nelghborhoods at the origin, for
to
the
topology
condition
Iff
7
of
every
to 7, converges with respect to
Iff
there
is
a
T;
basis
of
consisting of sets whlch are closed with respect
coordlnate-wlse
filter,
condlt[on
Cauchy
.
convergence
wlth
E; (E,) satisfies the filter
on
respect
to
,
convergent
with
respect
Examples are given of solid (deflnltion below) normed spaces of sequences which
(a) fall to satisfy the filter condition, or (b) satisfy the filter condition, but not
the
closed
fulfilling
neighborhood
(a),
(Robertson
condition.
others
and
have
given
examples
and examples fulfilllng (b), but these examples were not solid, normed
spaces.)
However, it is
pairs (E,T), the filter and closed
shown
sequence
that
neighborhood
among
separable
separated,
conditions
are
solid
and
equivalent,
equivalent to the usual coordlnate sequences constituting an unconditlonal Schauder
basis
for (E,T).
unconditional
pair
Consequently,
Schauder
basis
the
in
usual
every
coordinate
complete,
sequences
separable,
do
constitute
separated,
an
solid
(E,).
KEY WORDS AND PHRASES.
separated,
separable,
Topological vector space, sequence space, norm, Banach space,
solid,
filter
condition,
closed
neighborhood
condition,
unconditional Schauder basis, Abschnltt-Konvergenz (AK).
1980 AMS SUBJECT CLASSIFICATION CODE. 46A45.
BACKGROUND AND DEFINITIONS.
.
and
Suppose that E is a topological vector space under two topologles,
the
closed
satisfies
that
[I]
(E,)
shall
say
and Garllng [2], we
Following Robertson
neighborhood condition (with respect to 7) if and only if there is a base of
r-nelghborhoods of the origin which are -closed, and that (E,T) satisfies the filter
P.D. JOHNSON, JR.
626
(with respect to ) if and only if each filter in E, Cauchy with respect
condition
to T, and convergent with respect
,
to
converges with respect to
T.
(More exactly,
Robertson might say that (E,) satisfies the filter condition with respect to
(E,T)
and the identity injection of
Observe
that
(E,z)
if
is
condition, with respect to any
theorem
of
Robertson
(Hausdorff), and
The
.
complete,
.
importance
(E,)
of
(E,z) trivially
then
,
filter
I)
worth
is
noting:
if (E,) is
separated
then the completion of (E,T) is naturally embedded
(E,T)
if and only if
satisfies the filter condition, with
closed neighborhood condition arises
the
the
satisfies
Although the result will not be used much here, a
Theorem
is finer than
in the completion of
respect to
[I],
(E,).)
into
from a result of
([3], Proposition 8, Chap. i., I), which may also be found in Treves [4]
(Lemma 34.2); this result is approximately Proposition I0 of [I], which we restate
Bourbaki
here.
LEMMA
l_f (E,)
I.I.
is
separated,
closed neighborhood condition, then
T is
finer
than
(E,z) satisfies the
In this paper E will be a subspace of
,
,
and (E,) satisfies
the
filter condition.
the vector space of all scalar sequences
will be the topology of
(the scalars may be either the real or complex numbers);
This is the relative topology on E induced by the
coordlnate-wlse convergence on E.
product
topology on m, thought of as a countable product of copies of the scalar
field.
the solid hull of x is
If x E
A subset of
elements.
solid
is
if
and only if
it
Throughout, E will be solid.
contains
A topology
the solid hull of each of its
T
on E (with which E becomes a
t.v.s.) will be called solid if there is a neighborhood base at the origin for
are solid, we will refer to (E,z)
When both E and
consisting of solid sets.
as a solid
pair.
Note that if a norm p on E satlslfes
y
E
S(x)E implies that p(y)
(I.I)
p(x),
and E is solid, then the norm topology on E associated with p is solid.
Conversely,
if E is solid, any solid norm topology on E is associable with such a norm (since the
convex hull of a solid set is solid); when
satisfies (I.I)
will be called a solid
norm.
LEMMA 1.2.
denote the
Suppose that E is solid, and p is a solid norm on E. Let
}. Then (E, ) satisfies the
{x e E; p(x)
closed neighborhood condition if and only if the closure of U i__n_n (E,w)is bounded
topology on E defined by p, and let U
in
(E, ).
627
CLOSED NEIGHBORHOOD AND FILTER CONDITIONS IN SEQUENCE
If the closure W of U in (E,) is bounded in (E,), then {rW; r
PROOF.
a neighborhood base at the origin in
(E,) consisting of
sets closed in
the other hand, (E,)satslfes the closed neighborhood condtion,
in
(E,) of some set rU, r
>
0, is contained
in
U; thus
>
0} Is
(E,).
If, on
then the closure
Wc_r-Iu.
The coordinate projections on E are the functlonals f de[ined by
n
f (x) =x
n
n
The following lemma is well known.
A proof of the equivalence of (a) and (c) appears
in Johnson and Mohapatra [5] (Prop. 2.5).
Suppose that (E,) is a solid pair. The following are equivalent.
LEMMA 1.3.
The coordinate p.roJectlons are continuous on E.
(a)
(b)
is finer than
(E,)
(c)
The
coordinate
m, e (m)
7.
is separated.
sequences are
the sequences
en
e
defined
by
en(m)
We shall assume throughout that E contains the coordinate
E
E defined by
The finite sections are the functions P
n
Following the terminology in Kothe [6] if for each x e E the
0 otherwise.
n
sequences.
Y. xkek.
P (x)
kgn
sequence
(Pn(X))
converges to x in
When (E, ) is a
(E, ), we will say that (E, ) is AK (for Abschnitt-Konvergenz).
form a
sequences
separated solid pair, (E,)is AK If and only if the coordinate
then guarantees that the basis is
Schauder basis for (E,); the solidity of E and
unconditional.
The following is Proposition 2.9 of [5].
LEMMA 1.4.
Suppose that (E,) is an AK solid pair.
Then
(E,) satisfies the
closed neighborhood condition.
In the
next section we give two examples of solid, solidly normed spaces
the first fails to satisfy the filter condition, and the second satisfies the
E
filter condition, but not the closed neighborhood condition.
the supply of such examples;
The object is to augment
see Lindenstrauss and Tzafrlfi
[7], Garllng [2], and
A far as we know, these are the first known (or, in the
case of Example 2.1, noticed) such examples in which (E,x) is a solid, solidly normed
is the topology of coordlnate-wlse convergence on E.
sequence space, and
The norm topology in Example 2.1 is not consistent in the sense of Ruckle [9]; by
Gaposhkin and Kadets ([8]).
Theorem
of [2] It could not possibly be.
In section 3
The norm in Example 2.2 is consistent.
it is shown that among separable, separated solid pairs
(E,z), the
(E,)
filter and closed neighborhood conditions are equivalent, and are equivalent to
being
AK.
As an easy consequence,
it
follows
that
in
every complete separable
separated solid pair (E,), the coordinate sequences form an unconditional Schauder
(E,z). (Thus, the famous separable Banach spaces with no Schauder basis,
basis of
P.D. JOHNSON, JR.
628
while possibly realizable as
sequence spaces, are not realizable as solid,
solidly
normed sequence spaces.)
In Section 4 we discuss a problem mentioned in [5], still unresolved, related to
the results of this paper, and one other problem that arises from these results.
2.
EXAMPLES.
EXAMPLE 2.1.
; (nx)
n
{x
Let E
(.[x n 12) 1/2
p(x)
+ lira
is bounded}.
n
suplnXnl.
Let x
Clearly E is solid, and p is a solid norm on E.
x
x
(x(k)) k
Then
for all k;
(k)
n
n
(k)
-I
if n
k,
by Lemma 1.3, (x
thus,
each n, define
(E,), and O(x (k)) >
the norm topology
converge tn (E,’),
:
(k))
cannot
Consequently, (E,) does not satisfy the filter condition.
Partition the positive integers into infinite sets
.
n=l
O(x)
IXkl
k J
n
Pn(X) <
[.
+ n lira sup
n
} and define p on E by
Pn(X)"
,
Clearly E is solid and p is a solid norm on E; E is an
), each isomorphic
to
so E, with
,
For each n, let
gn
Jn’
0 otherwise.
and
O(Pk(gn ))
(E,) does
3.
gn(m)
"I
direct
T
for each k, and
gn
(gn
is the limit, in
the spaces
is the topology determined by p.
denote the characteristic sequence of
Note that
sum" of
Consequently, (E,)
is complete.
satisfies the filter condition, where, as usual,
m
For
Ji,J2,
by
on
Pn
Pn(X)--sup
keJ
Jn
e E be defined by
is Cauchy with respect to p, converges to zero in
EXAMPLE 2.2.
{x
(k)
0if n <k.
associated with p.
Let E
Let p be defined by
n +
(E,),
Jn;
i.e.
for each n.
of
(Pk(gn))k
gn(m)
if
Also,
By Lemma 1.2,
not satisfy the closed neighborhood condition.
ON SEPARABLE SOLID PAIRS.
THEOREM 3.1.
Suppose that (E,) is a separable,
separated solid pair.
The
following are equivalent.
(E,) satisfies the filter condition.
(E,) satisfies the closed neighborhood condition.
(c)
(E, I) is AK.
PROOF. By Lemma 1.3 and I.I, (b) implies (a), and (c) implies (b) by Lemma 1.4.
(a)
(b)
629
CLOSED NEIGHBORHOOD AND FILTER CONDITIONS IN SEQUENCE
What remains to be shown is that (a) implies (c).
Suppose that (E,x) is not AK. Then there is some x E E such that the sequence
(P (x)) does not converge to x in (E x). But (P (x)) does converge to x In (E,);
n
n
therefore, if (a) holds, it must be that (P (x)) is not Cauchy in (E,x). Therefore,
n
there is a solid T-nelghborhood U of the origin in E and two sequences
(nt),
t)
of
U
is
(m
positive integers satisfying
n
<
t
m
nt+
t
and
P
If t
<
m
(x)- P
n
t
r then n
<
t
m
(x)
nr,
t
Pn r (x) Pnt(X)
consequently,
U,
t
so
Pint(x) Pnt(X)
1,2,
t
U for
>
r
t,
e S(P
n
(x)
P
r
because P
mt
(x)
(x));
n
P
t
nt
(x)
U and
solid.
For each t, let
B
{n
t
t
+
n
t+1
}"
For each set T of positive integers, let
yn(T)=
Y(T)
n
Each
y(T)
x if n s
n
y(T)E
be defined by
UB t’
tT
0 otherwise.
is in the solid hull of x, so
y(T)
E for each T.
(T
(T
sets of positive integers, then the solid hull of y
y
(T
P
nt+
(x)
P
n
Suppose
(x)
U, for some t; consequently y
(T
y
2
T
If T
2
are two
contains
2
t
V
is
satisfying V + VU.
positive Integers
open T-nelghborhood of
balanced,
The paragraph preceding Implies that
a
the
{y(T)+
origin
V; T is a
in
E
set
of
is an uncountable collection of mutually disjoint r-open non-empty
subsets of E; consequently
assuming (a) and not (c).
(E,x)
is not separable.
Consequently,
We have arlved at this conclusion
(a) implies (c), when (E,x) satisfies the
hypothesis of the theorem.
COROLLARY
then
(E,x)
PROOF.
3.1.
If (E,x) is
a
complete
separable,
separated
solid
pair,
is AK.
If (E,x) is complete, then (E,x) satisfies the filter condition.
It has been pointed out to the author that the hard part of the theorem above,
(a) implies (c), resembles a corollary of a well known result about Banach lattices
([7], Proposition l.a.7). Since the scalars may be the complex numbers here, it would
P.D. JOHNSON, JR.
630
be reckless to deduce the theorem from the Banach lattice result, even in the special
(E,T)
case in whlch
to
is a Banach space, but the point is well made; the author admits
rediscovered
having
an argument
and
a
result
already
in
the
literature,
in a
The author is of the opinion that such rediscoveries, if
somewhat different settlng.
recognized as such, do no harm, and may serve to stimulate interest in the original.
4.
REMARKS AND PROBLEMS.
Suppose (E,T) is
a
solid
pair
negative entries and no zero columns.
[5, I0, II],
(a
and A
is
mn
an
infinite
matrix
with
non-
Followlng the notation of Johnson and Mohapatra
we set
nor-A-l(E)
{x
:
Alx
(
n
amnlXnl)m
E}.
-I (E) is
a solid subspace of ; if we require each column of A to be in E,
then nor-A (E) contains the coordinate sequences; further, nor-A (E) is naturally
-I
t.v.s,
topology, nor-A (T), obtained by taking as a
equipped with a solid
Then nor-A
-I
neighborhood base at the origin, the sets
nor-A-l(u)
{x e w;
A[x[
U},
as U ranges over solid T-neighborhoods of the origin in E.
-!
-I
[5] it is shown that (nor-A (E), nor-A ()) will inherit a great deal
from (E,z) (the property of being AK, for instance), but there seems to be a
In [5] (Theorem 2.13) and [II] (Prop.
mysterious difficulty regarding completeness.
nor-A-l()) is complete when (E,z) is, provided
I.I) it is shown that
In
(nor-A-l(E),
either that
(E,T)
satisfies the closed neighborhood condition, or that the rows of A
(In [II], A is lower triangular, but this condition can be
Richard Haydon and Mireille Levy have since shown the author, in a private
communication, that these provisions can be omitted when (E,T) is a Banach space, and
In
their proof extends to the cases in which (E,T) is locally convex and metrizable.
are finite sequences.
relaxed.)
general, however, the problem remains unresolved.
PROBLEM 4.1.
Suppose (E,T) is a separated, complete solid pair
infinite matrix with non-negative entries, no zero columns, and
(nor-A -I (E), nor-A
and A is an
every column
in E.
l__s
(T)) necessarily complete?
The remarks preceding say that if a counterexample can be found, then A will have
to
have
some
entries, and
rows
(indeed,
infinitely
many
rows) with infinitely many non-zero
(E,) will not be locally convex and metrizable.
It is shown in [5] (Theorem 2.10) that if (E,T) satisfies the closed neighborhood
-I
-I
condition, then so does (nor-A (E), nor-A
Suppose that (E,z) and A are as in Problem 4.1, except that (E,T)
is not necessarily compl@te.. Suppose that (E,z) satisfies the filter condition. Does
PROBLEM 4.2.
CLOSED NEIGHBORHOOD AND FILTER CONDITIONS IN SEQUENCE
(nor-A
-I
(E),
nor-A
-I
631
()) necessarily satisfy the filter condition?
Problems 4.1 and 4.2 are equivalent in the following sense.
Suppose that C is a
class of separated solid pairs, with the property that if (E,T) c C and (E,)
(E,)of (E,)also
satisfies the fiter condition, then the completion
C.
of
(By Robertson’s
straightforward
([I]), (E,)
Theorem
(,)
that
see
to
will
be
be
will
solid
a
a
pair).
is an element
sequence
space;
the
answer
If
it
to
is
the
question in 4.1 or 4.2 is yes, whenever (E,T) is confined to C, then the answer to the
whenever (E,) is in C (with A fixed, or allowed to
question in the other is yes,
range over some selection of matrices).
Thus, for instance, the result of Haydon and
mentioned above implies that when (E,) Is locally convex, metrlzable, and
satlfles the filter condition, and A satisfies the hypothesis of Problem 4.1 with
respect to(E,), then (nor-A (E), nor-A ()) sat[sfles the filter condition.
For
Levy
another
instance,
the
answer
to
4.2
whenever
yes
is
the
of
rows
A are
finite
sequences.
We omit the proof of the equivalence.
In one direction the proof uses the
theorem of Robertson mentioned above, and the fact that the completion of a solid pair
s
a solid
pair,
when it
In the other direction the proof
space.
a sequence
is
resembles that of Theorem 2.13 of [5].
Finally, we note a problem left unresolved by the results of sections 2 and 3.
Does there exist a separable, separated solid pair (E,), preferabl
PROBLEM 4.3.
normed, which does not satisfy the filter condition?
To see this, let
The space in Example 2.1 is not separable.
Dr;
r c c} be a
collection of infinite sets of positive integers indexed by the continuum c with the
property
that r
r
that K
2 implies
the
r
0 K
r
is
obtained
by
rational
numbers, and for each real number r,
sequence
of
putting
distinct
2.1,
(r
p(x
{x(r):
(r
x
rationals
1/n if n
defined by x
positive
K
integers
convergent
x(r)
n
r’
(Such a collection may be
finite.
2
in
to
one-to-one
taking K
r.)
0 otherwise.
r
correspondence
to be
with
the
the elements of a
For each r
c, let x(r) e be
Then, with E and p as in Example
r E e} is an uncountable collection of elements of E, with
2
)
for r
r
2.
The theorem of section 3 bears on Problem 4.3; it suffices to find a separable,
separated solld pair
(E,z) which
each x E E, the sequence
(Pn(X))
is not AK.
The proof of that theorem says more; for
will have to be Cauchy.
ACKNOWLEDGEMENT.
Besides Richard Raydon and Mireille Levy, whose invaluable (to me) contribution
is
mentioned
earlier
in
this
section,
I would like to thank R.N. Mohapatra, Stan
Rajnak, and M.S. Ramanujan for various helpful and illuminating remarks on the subject
of this paper.
632
P.D. JOHNSON, JR.
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