In 1997, J. A. Fridy gave conditions for noninclusion of
ordinary and of absolute summability domains. In the present note,
these conditions are interpreted in a natural topological context
thus giving new proofs and also explaining why one of these
conditions is too weak. Also an open question posed in Fridy's paper is
answered

W. Beekmann

International Journal of Mathematics and Mathematical Sciences

English

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Internat. J. Math. & Math. Sci.
Vol. 23, No. 2 (2000) 119–122
S0161171200001861
© Hindawi Publishing Corp.
NONINCLUSION THEOREMS:
SOME REMARKS ON A PAPER BY J. A. FRIDY
W. BEEKMANN
(Received 19 August 1998)
Abstract. In 1997, J. A. Fridy gave conditions for noninclusion of ordinary and of absolute
summability domains. In the present note, these conditions are interpreted in a natural
topological context thus giving new proofs and also explaining why one of these conditions
is too weak. Also an open question posed in Fridy’s paper is answered.
Keywords and phrases. Summability matrix, summability domains inclusion, absolute
summability domains inclusion.
2000 Mathematics Subject Classiﬁcation. Primary 40D25, 40D09, 40C05.
1. Noninclusion for ordinary summability. Recently, J. A. Fridy [2] stated a non-
inclusion theorem that can be formulated in the following way.
Theorem 1.1. Let A and B be regular matrices such that cA, the summability do-
main of A, is included in cB , the summability domain of B. Then
lim
n,k
ank = 0 ⇒ lim
n,k
bnk = 0. (1.1)
Here limn,k ank = 0 (and, similarly, limn,k bnk = 0) is taken in the Pringsheim sense,
that is,
∀
 > 0∃N > 0 : (n>N and k >N) ⇒ ∣∣ank∣∣< 
. (1.2)
Of course, this is a noninclusion theorem, since if A has that limit property and B
does not, then cA  cB . The reason for the above formulation is that it emphasizes an
invariance property which is stated in an invariant form in the Lemma 1.2. Therein,
ek denotes the basic sequence ek = (0, . . . ,0,1,0, . . .) with “1” in the kth position, and
the summability domain
cA =

x = (xk)|Ax =
( ∞∑
k=1
ankxk
)
n=1,2,...
exists and converges

 (1.3)
is endowed with its FK-topology (see, e.g., [3, Ch. 22]) which is given by the seminorms
pr (x) := |xr | (r = 1,2, . . .),
qr (x) := sup
m
∣∣∣∣∣
m∑
k=1
arkxk
∣∣∣∣∣ (r = 1,2, . . .),
p0(x) := ‖Ax‖∞ = sup
n
∣∣∣∣∣
∞∑
k=1
ankxk
∣∣∣∣∣.
(1.4)
120 W. BEEKMANN
Observe that all column limits of A exist if and only if ϕ := span {e1,e2, . . .} ⊂ cA.
Lemma 1.2. Let A be a matrix with existing column limits. Then(
lim
k→∞
ank = 0 for n= 1,2, . . . and lim
n,k
ank = 0
)
⇐⇒ lim
k→∞
ek = 0 in cA. (1.5)
Proof. Certainly, pr (ek)→ 0 as k→∞ for each r . Also, the condition limk→∞ank =
0 for n = 1,2, . . . (all row limits of A are zero) is equivalent to limk→∞qr (ek) = 0 for
r = 1,2, . . . . Now, let limn,k ank = 0 in the Pringsheim sense. Then, given 
 > 0, there
exists N1 > 0 such that |ank|< 
 for n>N1 and k >N1. If, in addition, limk→∞ark = 0
for r = 1, . . . ,N1, then there exists N > N1 such that |ank| < 
 for 1 ≤ r ≤ N1 and all
k > N . Thus p0(ek) = supn |ank| ≤ 
 for all k > N . Hence p0(ek) → 0 as k → ∞, and
ek→ 0 in cA follows.
Conversely, suppose ek → 0 in cA. Then, in particular, limk→∞qr (ek) = 0 for r =
1,2, . . . and p0(ek) = supn |ank| → 0 as k→∞; the former implies limk→∞ark = 0, the
latter limn,k ank = 0.
As a corollary we obtain Fridy’s result.
Corollary 1.3. Let A be a matrix with existing column limits and with row limits
zero. If cA ⊂ cB , then
lim
n,k
ank = 0 ⇒ lim
n,k
bnk = 0, (1.6)
and then, in fact, B is a matrix with existing column limits and with row limits zero.
Proof. By the Lemma 1.2 we have ek → 0 in cA. By cA ⊂ cB , the relative topology
of cB on cA is weaker than the FK-topology of cA (see [3, Ch. 17]; hence ek → 0 in cB ,
and, by Lemma 1.2, this means limn,k bnk = 0, and the row limits of B are zero.
Remark 1.4. In [2] it is already noticed that in Theorem 1.1 the supposition that
A and B should be regular can be relaxed to the condition that both matrices have
column and row limits zero. Corollary 1.3 is slightly more general; the existence of the
column limits of A is needed in order that ek ∈ cA for all k, and hence, by cA ⊂ cB , the
column limits of B exist. It should also be remarked here that a K-space E containing
ϕ is called a wedge space if ek→ 0 in E, see G. Bennett [1, Thm. 27], asserting that cA
with ϕ ⊂ cA is a wedge space if and only if limk→∞ supn |ank| = 0.
2. Noninclusion for absolute summability. In [2] noninclusion is also considered
for absolute summability; here
A =

x = (xk)
∣∣∣∣Ax =
( ∞∑
k=1
ankxk
)
exists and Ax ∈ 

 (2.1)
the absolute summability domain of A, is concerned, where
 =
{
x = (xk) | ‖x‖1 :=
∞∑
k=1
|xk|<∞
}
. (2.2)
We state the result in the following form.
NONINCLUSION THEOREMS: SOME REMARKS . . . 121
Theorem 2.1. Let A be a matrix with its column sequences in  (so that ek ∈ A for
all k), and let B be a matrix with A ⊂ B . If there is an index sequence (k(j))j=1,2,...
such that
lim
j→∞
∞∑
n=1
∣∣an,k(j)∣∣= 0, (2.3)
then
lim
j→∞
∞∑
n=1
∣∣bn,k(j)∣∣= 0. (2.4)
In [2], there is an extra condition  ⊂ A, but condition (2.3) is relaxed to
lim
j→∞
∞∑
n=µ
∣∣an,k(j)∣∣= 0 for some integer µ, (2.5)
and (2.4) is correspondingly weakened to
lim
j→∞
∞∑
n=µ
∣∣bn,k(j)∣∣= 0 (2.6)
with the same µ as in (2.5). Unfortunately, this relaxed version fails for µ > 1, even if
 ⊂ A and the µ in (2.6) is allowed to diﬀer from that one in (2.5). This can be seen
from the following example.
Example 2.2. For all k= 1,2, . . . , deﬁne
ank :=

1, if n= 1,0, if n> 1, and bnk :=
1
n2
for n= 1,2, . . . , (2.7)
so that
(Ax)n =


∑∞
k=1xk, if n= 1,
0, if n> 1,
and (Bx)n = 1n2
∞∑
k=1
xk. (2.8)
Then, clearly,
 ⊂ A = B =
{
(xk)
∣∣∣∣
∞∑
k=1
xk converges
}
,
lim
j→∞
∞∑
n=2
∣∣an,k(j)∣∣= 0, lim
j→∞
∞∑
n=µ
∣∣bn,k(j)∣∣=
∞∑
n=µ
1
n2
> 0
(2.9)
for each integer µ and each index sequence (k(j)).
To prove Theorem 2.1 in a topological way—similar to the proof of Corollary 1.3
(and Theorem 1.1)—we need the following lemma.
Lemma 2.3. Let A be a matrix with its column sequences in , and let (k(j))j=1,2,...
be an index sequence. Then
lim
j→∞
∞∑
n=1
∣∣an,k(j)∣∣= 0⇐⇒ ek(j) → 0 in A as j →∞. (2.10)
Proof. The FK-topology of the FK-space A is given by the seminorms pr ,qr (see
above) and
122 W. BEEKMANN
p0(x) := ‖Ax‖1 =
∞∑
n=1
∣∣∣∣∣
∞∑
k=1
ankxk
∣∣∣∣∣. (2.11)
Thus ek(j) → 0 in A is equivalent to pr (ek(j)) → 0,qr (ek(j)) → 0 for each ﬁxed r =
1,2, . . . and ‖Aek(j)‖1 =
∑∞
n=1 |an,k(j)| → 0. These conditions are equivalent to the
single condition ‖Aek(j)‖1 → 0, since qr (ek(j)) ≤ ‖Aek(j)‖1 and prH(ek(j)) = 0 for
k(j) > r . The lemma follows.
Theorem 2.1 is now a simple corollary of Lemma 2.3. By A ⊂ B the FK-topology of
A is stronger than the relative FK-topology of B on A. Hence ek(j)→ 0 in A implies
ek(j)→ 0 in B . Lemma 2.3 now yields the assertion of Theorem 2.1.
In [2] it is asked whether in Theorem 2.1 conditions (2.3) and (2.4) can be replaced
by
lim
j→∞
∣∣∣∣∣
∞∑
n=1
an,k(j)
∣∣∣∣∣= 0 and limj→∞
∣∣∣∣∣
∞∑
n=1
bn,k(j)
∣∣∣∣∣= 0, (2.12)
respectively. The answer is negative as can be seen by the following example.
Example 2.4. Deﬁne A= (ank) and B = (bnk) by
ank :=


1, if n= 1,
−1, if n= 2, for k= 1,2, . . .
0, if n> 2,
(2.13)
and
bnk :=

1, if n= 1,0, if n> 1, for k= 1,2, . . . , (2.14)
so that Ax = (∑∞k=1xk,−∑∞k=1xk,0,0, . . .) and Bx = (∑∞k=1xk,0,0, . . .).
Then, clearly, ( ⊂)A = B = {x = (xk)|
∑∞
k=1xk converges} and
lim
j→∞
∣∣∣∣∣
∞∑
n=1
an,k(j)
∣∣∣∣∣= 0, limj→∞
∣∣∣∣∣
∞∑
n=1
bn,k(j)
∣∣∣∣∣= 1 (2.15)
for any index sequence (k(j))j=1,2,....
References
[1] G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine
Angew. Math. 266 (1974), 49–75. MR 49 9585. Zbl 277.46012.
[2] J. A. Fridy, Noninclusion theorems for summability matrices, Internat. J. Math. Math. Sci.
20 (1997), no. 3, 511–516. MR 98f:40003. Zbl 886.40005.
[3] K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren, 2nd ed., Ergebnisse der
Mathematik und ihrer Grenzgebiete, vol. 15, Springer-Verlag, Berlin, NewYork, 1970.
MR 41#8863. Zbl 199.11301.
Beekmann: Fachbereich Mathematik, Fern Universität-Gesamthochschule, D 58084,
Hagen, Germany
E-mail address: wolfgang.beekmann@fernuni-hagen.de
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Noninclusion theorems: some remarks on a paper by J. A. Fridy

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