ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE X X (1978)
K g P e n g - N u n g and L e e P e n g - Y e e (Singapore)
1. Introduction. In [1] Jagers has determined the Kôthe duals of the Cesàro sequence spaces ces^, 1 < p < oo, where cesp consist of all real sequences x = {xk)keN such th a t
The case when p — 1 is trivial (see [4]). The case when p = oo has been considered by the present authors [3]. In this note, we shall define the Cesàro sequence spaces in a different way, th a t is, with norms \\x\\p which do"not satisfy the absolute property: = \\x\\p , where \x\ = (|%|)fc6iv>
and try to determine their associate spaces, i.e., the Kôthe duals.
2. Normed Kôthe sequence spaces. Let X be the set of all real se
quences x = {xk)keN. A functional q from X into the non-negative extended real number system is called a sem i-norm if
(i) e(0) = 0, (ii) q(ox) = \a\q(x), (iii) Q{x + y ) ^ Q(x) + Q{y).
If instead of (i) q satisfies the condition th a t q{x) = 0 if and only if x = 0, then q is called a norm . We denote by X e the collection of all sequences x satisfying q{x) < oo. Obviously X e is a linear space, and we call X Q the norm ed K ôth e sequence space of non-absolute type with the semi-norm q. If X Q is complete with respect to the norm q, then X e is called a Banach sequence space of non-absolute type since we did not assume the absolute property.
From now on, let us assume th a t X e is a Banach sequence space of non-absolute type. Given a semi-norm q, we define a new semi-norm o'
as follows:
Cesàro sequence spaces of non-absolute type
oo
430 Ng P e n g -N u n g and Lee P e n g -Y e e
and we put q' (x) = o o i f the series JT xkyk does not converge for some y A=1
satisfying g(2/ ) < l . The semi-norm q' is called the associate semi-norm of q. The space X Q- consisting of all sequences x e X with q (x) < с о is called the associate space of X e. For any x = {xk)keN e X Q and у = (yk)keN e X Q we have
CO
\£хкУк\ < ?(*)$'(?)•
A = 1
A semi-norm q is said to be saturated if for every non-empty subset JE of N, there exists a non-empty subset F of E such th a t g(xF) < со,
where the sequence xF = (xk)keN is defined as xk = 1 if Te e F and xk = 0 if It $ F. I t is easy to see th a t q is saturated if and only if X Q contains all sequences of finite terms, i.e., sequences having only finitely many non-zero terms. The following theorem is a consequence of the B anaeh- Steinhaus theorem:
Th e o r e m 2 . 1 . Let q' be saturated, and y e X . Then y e X e. i f and
OQ
only i f £ x k yk < oo for all x e X e.
k = 1
3 . Cesàro sequence spaces. Let X p ( 1 < p < с о ) and X œ be respecti
vely the spaces of all x e X with
and
for 1 < p < с о ,
n g N < OO.
Note th a t the above norms are saturated except for p = 1.
Th e o r e m 3.1. The space X p (1 < p < со) is a Banach sequence space of non-absolute type.
P ro o f. Let (x^)ieN be a Cauchy sequence in X p so th a t for given e > 0 we have \\x ^ ~ x^\\p < e for all i, j > X(e). We write х(г) = {х$)ы м . Then for fixed h, {a$){eN converges. If lim x $ = xK, then
i-*oo
for i ^ N(e) and all m.
Hence letting m->oo we have \\х^ — x\\p < e for all г > A(e). This proves th a t X p is complete for 1 < p < oo. The completeness of follows similarly.
Cesàrо sequence spaces 431 Theorem 3.2. cesp <= X p for 1 ^ p < oo and cesp Ф X p .
P ro o f. The inclusion is trivial. To see th a t cesp ф X p, we define a sequence (xk)keN by х г — —1 and xk = 2 ( —l ) fe, h = 2, 3, ... which is an element of X p bu t not of cesp for 1 < p < o o . Further, the sequence (xk)keat defined by xk = fc( —l ) fc is a member of Х ж b u t not of ces^. Finally the sequence {xk)keN defined by x t = —1 and xk = ( —l) fc (2fc —1) /7Ь(7c —1) for h = 2, 3, ... is an element of b u t not of cesl .
Theorem3.3. 1 < p < oo and a be defined on X p by o(x) = (o'n(a?))neiVr 1 n
where ffn(x) — — / Ф n k=i
Then a is an one-to-one bounded linear transfor
mation from X p onto the sequence spaee lp with operator norm 1.
The proof is easy- The result will be used in an essential way in a proof in the next section.
4. The associate space of X p . Let Yq be the space of all sequences у e X such th a t
(1) \кУк\ ^ AT for all к e X , OO
(2) ?.q(y) = ( 2 \ к ( у к - у к+1)\а)119< OO for 1 < q < o o ,
&=i
and Aw(y) = su p {\h(yk - Ук+г)\', h e N} < o o .
We shall show th a t Y a is the associate space X'p of X p , where 1 (рф 4-1 lq = 1 and with Xq — ||*||p the associate norm of
Lemma 4.1. I f у — (yk)keN is an element in the associate space X'p of X p, then the sequence (hyk)keN is bounded. I n particular, when p = o o , hyk->0 as &->oo.
P ro o f. Let у = {yk)keN be an element of X'p . Then ^ x kyk < o o
for every x = (xk)keN in X p . This implies th a t xkyk->0 as &->oo. Since *=i x = (jfc( — l)*)*ejv is an element of J M, we have hyk->0 as ~k->oo. In general, write s. 1 VI
fc=l
and we have
(A) yjfcCA»* —(fc -l)e A_ j]->0 as Tc^oo
for all sequences (sk)keN in lp by Theorem 3.3. So (A) is also true for all sequences ( ( - l ) k \sk\)keN in lp, i.e.,
( — [ *|e*l + ( *— as &-^oo.
I t follows th a t Jcyksk-±0 as Tc-+oo. Now we claim th a t (hyk)keN is bounded.
For p = 1, it is easy. Suppose th a t the sequence is not bounded for 1 < p
< o o . Then there exists a subsequence (Цук.^бМ such th a t \Цук.\> j for j e X. Take r such th a t 0 < r < 1 and pr > 1, and define a sequence
432 N g P e n g -N u n g and Lee P e n g -Y e e
{ * ' k ) k e N in l P ЪУ
, (tyk)~r ^Ьеп к = ICj, j = 1, 2, 3, ..., SJç ----
0 otherwise.
Then we have Цук.*к. = (ЦУк-)1 r <l°es not tend to zero as j-> oо which leads to a contradiction. Therefore the sequence must be bounded.
Theorem 4.2. The associate space X p of X p (1 < p < oo) is the space Y q with the norm l q, where 1 /p + l/g = 1.
O O
P ro o f. Let у e X p . Then ] ? xkyk is convergent for all x e X p . hfow
k = 1
we apply Abel transformation and obtain
Л ^ к У к = £ +
fc=i A = 1
where an = — I xk . Since x e X p , it follows th a t an->0 as n-> oo for
*=i
1 < P < 00 and th a t (<rn)neN is bounded for p = o o . In view of Lemma 4.1, the last term in the above equality tends to zero as m~>oо and hence
£а>кУк = ]>]ЧУк-Ук+1)<*к-
/с=i *=i
Therefore applying Theorem 3.3 we see th a t the above series converges for all sequences a = (ak)keN belonging to lp . I t is known th a t (k (yk — 1) ) ^ belongs to lq, where 1 /p + l/q = 1 and th a t *
K(y) =
k = 1
O O
= sup {| £ xk ykI ; \\cc\\p < l} = ||y Hi.
*=i
Hence, together with Lemma 4.1, we have proved th a t X'p c Y q.
Conversely, let у — {yk)keN be an element in Y g. Since the norm Aq is saturated, then by Theorem 2.1 and Abel transformation we need only to prove th a t mykak->0 as &->oo for у e Y q and x e X p . The case when l < p < oo follows easily from the facts th a t {kyk)keN is bounded and <rfc->0 as Jc->oo. For p = oo, {ok)keN is bounded and lim yk = 0. I t
follows th a t k^°°
O O o o
Ы < ^ ь\Уз- уш \ < ^ j \ y j - y j+1\.
j— k j = к
Cesàro sequence spaces 433 By letting ft->oo, we obtain th a t ~kyk-^- 0 as fc ^ o o . This completes the proof.
We remark th a t in defining X p , the lp norm involved may be replaced by Or liez or Luxemburg norm for an Orliez sequence space. Hence we have defined an Orliez space of non-absolute type. Its associate space can also be found. The details, together with its general theory, are being worked out and will appear in a separate paper.
R e f e r e n c e s
[1] A. A. J a g e r s , A note on Cesàro sequence spaces, Nienw Arch, voor Wiskunde (3) 22 (1974), p. 113-124.
[2] G. M. L e ib o w itz , A note on Cesàro sequence spaces, Tamkang J. Math. 2 (1971), p. 152-167.
[3] Ng P e n g -N u n g and Lee P e n g - Y ее, On the associate spaces of Cesàro sequence spaces, Nanta Math. 9 (1976).
[4] J. S. S h iu e , On the Cesàro sequence spaces, Tamkang J. Math. 2 (1970), p. 19-25.
[5] A. C. Z a a n en , Integration, Chapter 15, Amsterdam 1967.
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