ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIX (1990)
H. Ca k a l l i* and B. Th o r p e (Birmingham)
On summability in topological groups and a theorem of D. L. Prullage
Abstract. In this paper, it is proved that the conditions for regularity given in [4] for summability methods defined on abelian topological groups are equivalent to those used in the classical case, i.e. regular methods are those which transform convergent sequences to convergent sequences leaving limit invariant.
1. Introduction. In [4] Prullage gave the definition of a limitation method in topological groups and introduced a concept of regularity involving Cauchy sequences. He continued in his papers [5], [6] and [7] (see also Maddox [3]
and Cakar [1] who studied summability in incomplete normed spaces).
The purpose of this paper is to prove that the conditions for regularity given in [4], are equivalent to those used in the classical case, i.e. regular methods are those which transform convergent sequences to convergent sequences leaving limit invariant.
2. Definitions and notation. By X, we will denote an abelian topological Hausdorff group, written additively, which satisfies the first axiom of count- ability. For a subset A of X, s(T) will denote the class of all sequences (x(n)) such that x(n)e A for n = 1, 2, ...; c(X), c0(X) and C(X) will denote the set of all convergent sequences, the set of all null sequences and the set of all Cauchy sequences in X, respectively.
In [4] Prullage gave the following definitions:
De f in it io n 1. A limitation method on X is a sequence of functions /(m) whose domain, for each m, is some subset of s (X) and contains the set С (X) and whose range is contained in X and satisfies that, if (x (н)), (у (и)) and their sum (x(n) + y(n)) are in the domain of f(m), then
f(m: x (1) + у (1), x (2) + у (2), ...) = f{nv. x(l), x(2), ...)+/(m : y(l), y(2), ...).
This research was supported by a grant from Erciyes University.
The intersection of the domains of the functions f(m ) will be called the domain of the limitation method ( f(m )).
Definition 2. A limitation method, (f(m )), is said to be triangular if it satisfies the following conditions:
(i) The domain of the method is all of s(X).
(ii) For every positive integer m and every pair of sequences (x(n)) and (y (n)) for which x (n) = y (n) for n = 1 ,..., m, it follows that
f(m: x (1), x (2 ),...) = /(m: y(l), y(2), ...).
In case (/(m)) is a triangular limitation method, we will write /(m:
x(l), x(2), ..., x(m)) instead of f(m : x(l), x(2), ...).
Definition 3. A limitation method on X, (/(m)), is said to be regular if, for each (x(n))eC (I),
lim (x(n)—f(m : x(l), x(2), ...)) = 0.
mfn~> oo
Definition4. If h is a continuous homomorphism from X into X, then the triangular limitation method, (/(m)), from X to X is said to be h-regular if for each Cauchy sequence (x(n)) of s(X),
lim (h(x(n))—f(m : x(l), x(2), ..., x(m))) = 0.
oo
Now we give the following definitions:
Definition5. A limitation method on X, (f(m )), is said to belong to (c(X), c(X), p) if
lim f(m: x(l), x (2),...) = lim x(n)
m~* oo n-+ oo
for each (x(n))ec(X).
The class (c0 (X), c0 (X)) will denote the set of all limitation methods which map each element of c0 (X) to an element of c0 (X). For the sake of simplicity, we write/instead of (f(m)) and/(x (n)) instead of (/(m: x (1), x (2),...)) when no confusion arises.
Definition 6. A limitation method (/(m)) from X to X is said to be generalized h-multiplicative if there exists a function h from X to X such that
lim /(m: x(l), x (2 ),...) = h(lim x (»))
m~* oo и-> oo
for all (x(w))ec(X).
3. Summability. Here, we prove a lemma analogous to one in [4], and then prove a theorem giving necessary and sufficient conditions for a triangular limitation method to belong to (c(X), c(X), p). It turns out that the
conditions are the same as those in Theorem 1 in [4], i.e. a triangular limitation method belongs to (<c(X), c(X), p) if and only if it is regular.
Lemma 1. Every topological Hausdorff group, which satisfies the first axiom of countability, has a base of closed neighbourhoods of the origin (see Theorem 3, page 46 in [2]).
Lemma 2. A limitation method on X, (f(m )), belongs to (с (X), c(X), p) if and only if the following conditions are satisfied:
(1) limm_ 00/(m: 0, 0 , . . . , 0, x, 0, 0 ,...) = 0 for all x in X in any fixed position.
(2) limm-.aofim: x, x , ..., x , ...) = x for all x in X.
(3) For each neighbourhood U of 0, there is a neighbourhood V of 0 such that for all (x(n))ec(T)ns(K ), there is an integer N such that m > N implies that f(m: x(l), x(2), ..., x(m), ...) is in U.
P ro o f. Sufficiency. Let (x(n)) be in c(X), and x(n) = l, say. Let W be any neighbourhood of 0. We may choose a neighbourhood 17 of 0 such that u + U + U c W.
By (3), there is a neighbourhood V of О such that for all (x (n ))ec(I)n s(K ), there is an integer N such that m > N implies that f(m : x (1), x (2 ),..., x (m),...) e U. Since lim„_* œ x (n) = l, there exists an integer M such that n ^ M implies that x(n) — leV. Then the sequence (0, 0 , . . . , 0, x(M) — l, x (M + 1) — /, x(M + 2) — l, ...) is in c (X )n s(F ) (indeed in c0(2i)ns(F)) and so there exists an integer M (l) > M such that m > M ( 1) implies that
f(m: 0, 0, ..., 0, x { M ) - l , x(M + l ) - / , x{M+ 2 ) - l , ...) e U .
By condition (1), there is an integer M(2) > M (l) so that if m > M (2), then /(m: x (1) — l, x (2) — l , ..., x (M — 1) — l, 0, 0, ...)g17.
Finally, from condition (2) there exists an integer M (3) > M (2) such that if m > M (3), then
(/(m: /, I...I, ...)—l)eU.
Thus if (3), then
f(m: x (1), x (2 ),. . . ) - / = [/(m: /, /, ..., / , ...) - l ]
+/(m: x (1) — /, x(2) —/ , ..., x ( M —l) — l, 0, 0, ...) +/(m: 0, 0 ,..., 0, x (M) — /, x(M + l ) - / , ...) e 17+17+17 c W.
Thus (f(m))e(c(X), c(X), p).
Necessity. The necessity of (1) and (2) is obvious. To prove the necessity of (3), let us take any neighbourhood U of 0.
By Lemma 1, we may choose a closed neighbourhood V of 0 such that V+Vc-U. Let (x (/)) e e (2Q n s (F ) and suppose that lim ,.^ x(j) = l. Since (f(m))e(c(X), c(X), p), there is an integer N such that m > N implies that
(/(m: x(l), x ( 2 ) , x ( m ) , ...)- l)e V . Then for m > N, it follows that
/(m: x(l), x ( 2 ) , x ( m ) , ...) = / + (/(m: x(l), x (2 ),...) — l)e V+Vcz U.
Thus condition (3) is necessary and the theorem is proved. ■ For triangular limitation methods we have the following result.
sTheorem 3. A triangular limitation method, (/(m)), belongs to (c(X), c(X), p) if and only if the following conditions are satisfied.
(Tl) lim m- 00/(m: 0, 0, ..., 0, x, 0, 0 ,. . . , 0) = 0 for all x in X in any fixed position.
(T2) limm_too/(m: x, x, ..., x) = x for all x in X.
(T3) For each neighbourhood U of O, there exists a neighbourhood V of О and a positive integer N such that if m and к are integers satisfying m > к > N and if x(n) is in V for n — k, fc+1, ..., m, then
f(m: 0, 0, ..., 0, x(k), x(/c+l), ..., x(m))et/.
P ro o f. Sufficiency. It is clear that conditions (Tl) and (T2) imply conditions (1) and (2) of Lemma 2. Let W be any neighbourhood of O. Choose a neighbourhood U of О such that U + U c W. By condition (T3), there is a neighbourhood F of О and an integer N such that m > k > N and у (n) in V(n = k, k + 1, ..., m) imply that
f(m: 0, 0, ..., 0, y(k), y { k + 1), ..., y(m))eU.
Let (x(n))Gc(X)ns(k). From condition (T3) there exists an integer M such that m > k > M and x(n) in V (n = к, k+ 1 ,..., m) imply that
f(m: 0, 0, 0, x(k), x(k-h 1), x(m))eU.
In particular, f(m: 0, 0, ..., 0, x(M + l), x(M + 2), ..., x(m))eU. From con
dition (Tl) there exists an integer N x > M such that if m > N lt then /(m: x(l), x(2), ..., x(M), 0, 0, . . 0)e 17.
So if m > N lt then
f(m: x(l), x(2) , . .., x(m)) =/(m : x (l), x(2), ..., x(M), 0, 0, ..., 0)
+f(m: 0, 0, ..., 0, x (M + l), x(M + 2 ),..., x(m)) eU + U <= W.
Thus condition (3) of Lemma 2 is also satisfied and it follows that (f(m))e(c(X), c(X), p).
Necessity. The necessity of (Tl) and (T2) is obvious. Let us prove that (T3) is also necessary. Suppose that (T3) is not satisfied so that there is a neigh
bourhood U of О such that for all neighbourhoods К of О and all integers N, there are integers m and к satisfying the condition m > k > N and there are elements y(n) in V for n = k, k + 1 , ..., m so that
J\m: 0, 0 , . . . , 0, y{k), y { k + 1), ..., у(т))фи.
Choose a neighbourhood W oi О such that W—W a U . Let V be any neighbourhood of O. Let (K(i))t® i be a sequence of neighbourhoods of О such that F(l) с V, and V(i+1) cz V(i) for i = 1,2, ... and П,® i V(i) = {0}.
Choose an integer m(l) and elements x(n) in V for n = 1, 2, ..., m(l) arbitrarily. Now suppose that an increasing sequence of integers m(i) (i = 1, 2, ..., r) and elements x(n) for n = 1 ,2 ,..., m(r) have been construct
ed. By condition (Tl) there is an integer n{r+ 1) > m(r) so that if к > n (r+ 1) then f(h. x(l)i x(2)...x(m(r)), 0, 0... 0)e Ж Also there exist integers m (r-fl) and k(r + 1) such that m (r+ 1) > fe(r+1) > n ( r + 1) and there exist elements x(n) in V(r) (n = k ( r + 1), k(r + 1 )+ 1 ,..., m (r+ 1)) so that
/(m (r+ l): 0, 0, ..., 0, x(k{r + \)), x(k{r + 1) + 1), ..., x(m {r+ 1)))£U.
Finally choose х(и) = 0 for n = m(r) + 1, m(r) + 2 , ..., k(r+ 1) — 1. Since
! V{i) = {0}, and x(n) is in V(r) for n — k (r + 1), Zc(r+1)+ 1, ... it follows that (x (n)) is a convergent sequence (in fact it is a null sequence). Let N be any integer. Choose an integer r such that m ( r + l) > N. Then
f( m ( r + 1): x(l), x{2), ..., x(m(r + l)))
-/( m (r+ l) : x(l), x(2), ..., x(m(r)), 0, 0, ..., 0)
= f(m{r + 1): 0, 0 , . . . , 0, x(k{r + 1)), x(fc(r+1) + 1 ),..., x(m(r + 1))).
Since the term on the right side of this equation is not contained in U, it follows that
/(m (r + l): x(l), x (2 ),..., х(т(г+ 1)))£Ж
Thus condition (3) of Lemma 2 does not hold. This contradiction completes the proof of necessity and hence the proof of the theorem.
We see below that the set of all triangular limitation method which belong to (c(2Q, c{X), p) is equal to the set of all triangular regular methods.
If we denote the set of all triangular limitation methods by T and the set of all regular limitation methods by R, then
( c ( X ) ,c ( X ) ,p ) n T = R n T . Prullage [4] proved the following theorem.
Th e o r e m 4. The triangular limitation method, ( f(m )), is regular if and only if the conditions (Tl), (T2) and (T3) of Theorem 3 are satisfied.
[This is Theorem 1 on page 261 in [4]].
Co r o l l a r y 5 . A triangular limitation method, (f(m)), belongs to (c(2Q, c(X), p) if and only if it is regular. That is
(< ( c ( X ) , c { X ) , p ) n T = R n T . Prullage ([4]) also gave the following lemma.
Le m m a 6 . A limitation method, ( / ( m ) ) , is regular if and only if conditions ( 1 )
and (2) of Lemma 2 hold and the following condition is satisfied:
(C3) For each neighbourhood U of О there is a neighbourhood V of О such that for all (x(n))eC (X )ns(F), there is an integer N such that m > N implies that f(m: x(l), x(2), ...) is in U.
[This is the Lemma on page 260 in [4]].
Co r o l l a r y 7 . A triangular limitation method, (f{m)), belongs to (c(X), c(X), p) if and only if conditions (Tl) and (T2) of Theorem 3 hold and condition (C3) of Lemma 6 is satisfied.
Le m m a 8 . A limitation method f = (f{m)) on X belongs to ( c 0 ( X ) , c 0 (2 Q ) if and only if it satisfies the condition (1) of Lemma 2 and the following condition:
(N3) For each neighbourhood U of О there is a neighbourhood V of О such that for all (x(«))ec0(2f) n. s(F), there is an integer N such that m > N implies that f(m: x(l), x ( 2 ) , x ( m ) , ...)eU.
P ro o f. Sufficiency. Let (x («)) e c0 (X). We must prove that (f(m: x(l), x(2) , . . . , x(m), ...)) is also a null sequence.
To prove this, let us take any neighbourhood W of 0. Then we may choose a neighbourhood U of О such that U + U <r W. By condition (N3), there is a neighbourhood V of О such that for all null sequences (y(n)) in s(F), there exists an integer N such that m > N implies that /(m: y(\), y(2), ..., y(m), ...)eU. Since (x(n)) is a null sequence, there is an integer N t such that x(n )e V for n ^ iV j. Then the sequence (0, 0 , . . . , x ( N t), x ( N 1 + l), ...) is a null sequence in s(F). So there exists an integer N 2 > iVA such that m > N 2 implies that
f(m : 0, 0 , . . . , 0, x (Nt), x {Nx +1), ..., x (m),...) e U.
On the other hand, by (1), there is an integer N 3 satisfying N 3 > N 2 such that f(m: x(l), x (2 ),..., x ( N 1 — 1), 0, 0, ...)eU for m > N 3.
Therefore for m > N 3
f(m: x (1), x(2), ..., x(m), ...) =f(m: x(l), x (2 ),..., x ^ - l ) , 0, 0 ,...) +/(m: 0 ,0 , . . „ O . x t N J . x ^ + l),...) e U + U c W.
It follows that (f(m : x(l), x ( 2 ) , x(m ),...)) converges to 0. Hence f e ( c 0(X), c0(X)).
Necessity. The necessity of (1) is clear. To prove the necessity of (N3), let us take any neighbourhood U of O. Choose a neighbourhood F of О such that F c U. Let (х(и)) be any null sequence in s(F). Since / belongs to (c0(2Q, c0 (X)), there is an integer N such that m > N implies that /(m: x(l), x(2 ) ,..., x(m), ...)eU. This completes the proof of necessity, hence
the proof of the theorem. В
Theorem 9. A triangular limitation method on X belongs to (c0 (X), c0(X)) if any only if it satisfies the conditions (Tl) and (T3) of Theorem 3.
P ro o f. Sufficiency. It is clear that condition (Tl) of Theorem 3 implies condition (1) of Lemma 2. Now, we are going to prove that condition (T3) implies condition (N3) of Lemma 8. Let W be any neighbourhood of 0. Choose a neighbourhood U of 0 such that U + U c W. By condition (T3), there is a neighbourhood F of О and an integer N such that m > k > N and у (и) in F (n = к, к 4- 1 ,..., m) imply that
f(m: 0, 0 ,. . . , О, y{k), y(k+ l), ...,y(m ))eU .
Let (x(w)) be any null (convergent or Cauchy) sequence in s(F). Then from condition (T3), there exists an integer M such that m > k > M and x(n) in F (n = к, k + 1 , ..., m) imply that
f(m: 0, 0, ..., 0, x(k), x(k + 1), ..., x(m))eU.
In particular
f(m : 0, 0 , . . . , x (M + l), x(M + 2 ),..., x{m))eU.
By condition (Tl), there exists an integer N t > M such that if m > N lt then f(m: x (1), x(2), ..., x(M), 0, 0 ,..., 0)et7.
So, if m > N t , then
f(m: x (1), x (2 ),..., x(m)) =/(m : x(l), x(2), ..., x(M), 0, 0, ..., 0)
+/(m: 0, 0 ,. . . , 0, x(M + l), x(M + 2 ),..., x(m)) e U + U <=W.
Thus condition (N3) is also satisfied, and it follows that f e ( c 0(X), c0{X)).
Necessity. The necessity of the condition (Tl) is obvious. The proof of the necessity of (T3) is similar to the proof of the necessity of (T3) in Theorem 3 and is omitted.
Corollary 10. Let (f{m)) be a triangular limitation method satisfying condition (Tl). Then the condition (T3) is equivalent to each of the conditions.
(N3), (3) and (C3).
P ro o f. This follows from the proofs of Corollary 7, Lemma 8 and Theorem 9. ■
Corollary 11. A triangular limitation method (f(m )) belongs to
(c0(X), c0(X)) if and only if it satisfies condition (Tl) and one of the conditions, (N3), (3), (C3) and (T3).
Theorem 12. A limitation method on X, (f(m)), is generalized h-multi- plicative if and only if it satisfies the condition (1) of Lemma 2, the condition (N3) of Lemma 8 and the following condition:
(H2) й т ^ ^ Д т : x, x, ..., x, ...) = h(x) for all x in X.
P ro o f. Sufficiency. Let (x(n))ec(2Q, and \imn_ ж x(n) = l, say. Let W be any neighbourhood of O. We may choose a neighbourhood U of 0 such that U + U + U cr W. By condition (N3), there is a neighbourhood V of 0 such that for all null sequences (x(n)) in s(F), there is an integer N such that m > N implies that
f(m: x (1), x(2), ...,x(m ), ...)eU .
Since lim,,.,^ x(n) = l, there is an integer M such that n ^ M implies that x(n) — leV. Then the sequence (0, 0, ..., 0, x(M) —l, x (M + l) — /, ...) is in с0(1)п5(К ), and so there exists an integer M (l) > M such that m > M (l) implies that
f(m: 0 ,0 , . ..,0 , x (M )~ l, x ( M + l ) - / , ...)eU .
By condition (1), there is an integer M (2) > M (1) such that if m > M (2), then f(m: x { \)~ l, x(2) — l , ..., x ( M - l ) - / , 0, 0, ...)eU .
Finally, by condition (H2), there exists an integer M (3) > M (2) such that if m ^ M (3) then
(/(m: /, l , . . . , l , . . . ) - h ( l ) ) e U . Thus if m ^ M (3), then
f(m: x (1), x (2 ),..., x(m ),...)-h{l) = (f{m: l, l, ..., I, ...)-h{l)) +f(m: x(l) — /, x(2) — / , ..., x ( M —l) — l, 0, 0, ...)
+/(m: 0, 0 , . . . , 0, x{M) — l, x(M + l ) - i , ...)e U + U + U c W.
Thus (/(m)) is a generalized h-multiplicative limitation method.
Necessity. Necessity of (H2) follows immediately. In fact, h may be defined as h(x) = limm_ 00/(m: x, x , ..., x , ...) for all x in X. So
h(x + y )= lim f(m: x + y, x + y , ..., x + y , ...)
m~+ oo
= lim (/(m: x, x, ...)+/(m : y, y , ...))
m~+ oo
= lim /(m: x, x, ...)+ lim f(m: y, y , ...) = h(x) + h(y)
m~* со wj-» oo
and therefore h(0) = 0. Hence f e ( c 0(X), c0(X)). Thus, by Lemma 8 , / satisfies the condition (1) of Lemma 2 and the condition (N3) of Lemma 8. This completes the proof of necessity, hence the proof of the theorem. ■
Theorem 13. A triangular limitation method on X , (f{m)), is generalized h-multiplicative if and only if it satisfies the conditions, (Tl) and (T3) of Theorem 3 and the following condition:
(TH2) limm_ 00/(m: x, x,‘..., x) = h{x) for all x in X.
P ro o f. It is clear that the conditions (Tl) and (TH2) are equivalent to the conditions (1) and (H2). Then, by Corollary 10, (T3) is equivalent to the condition (N3). So the conditions of this theorem are equivalent to those used in Theorem 12, and hence (/(m)) is generalized h-multiplicative if and only if it satisfies the conditions (Tl), (T3) and (TH2).
It should be noted from the proof of Theorem 12 that if a limitation method is generalized h-multiplicative, then h is an additive function on X.
Furthermore, h must also be continuous when (/(m)) is a generalized h-multiplicative triangular limitation method. ■
Theorem 14. Let (/(m)) be a generalized h-multiplicative triangular limita
tion method. Then h is additive and continuous, i.e. h is continuous homomor
phism.
P ro o f. Let (/(m)) be a generalized h-multiplicative triangular limitation method. T h e n /e (c 0(X), c0(X)). So by Corollary 1 0 ,/ satisfies the conditions (Tl) and (T3). From Corollary 11, it satisfies the condition (3) of Lemma 2. We can write h(x) = limwl_>00/(m: x, x, ..., x) for all x in X. Since h is additive, it is enough to show that it is continuous at 0. Let W be any neighbourhood of h (0) = 0. We may choose a closed neighbourhood U of О such that U cr W (by Lemma 1). Then, by condition (3), there is a neighbourhood V of О such that for all (x(n))ec(X )ns(F), there is an integer N such that m > N implies that f(m : x(l), x (2 ),..., x{m))eU. In particular for all xeV , there exists an integer M such that m > M implies that f(m: x, x, ..., x)eU. So, since U is closed, it follows that
h (x) = lim f(m: x, x , ..., x) e U a W.
oo
Thus for all x e V, h (x) e W. We have shown that for each neighbourhood W of О = h(0), there is a neighbourhood V of О such that h(V) cz W. It follows that h is continuous. ■
Prullage ([4]) gave the following theorem (page 270, Theorem 5):
Theorem 15. Let h be a continuous homomorphism from X to X. The triangular limitation method, (f{m)), is h-regular if and only if it satisfies the conditions of Theorem 13.
We thus have the following result:
Co r o l l a r y 16. A triangular limitation method from (/(m)), is generalized h-multiplicative if and only if it is h-regular.
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DEPARTAMENT O F MATHEMATICS, THE UNIVERSITY OF BIRMINGHAM, BIRMINGHAM, ENGLAND
