O n modular spaces oî strongly summable sequences

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X IX (1976)

J

A

(Bratislava)

O n modular spaces oî strongly summable sequences

Modular spaces of strongly summable sequences are investigated in papers [1] and [2]. Also some generalizations of some results of above papers are included.

Let T, T b, T 0, T f denote the space of all real sequences, the space of all bounded real sequences, the space of all real sequences which con­

verge to zero and the space of all real finite sequences, i.e. sequences x = {tv}, where % — 0 except a finite number of v. Let A = {amn) be such a regular summability method for which amn > 0 and no column consists of zeros only. Let Ф = {(pv} denote a sequence of 97 -function (non­

decreasing continuous function 9 ? defined for w > 0 for which 99 ( 0 ) = 0 ,

<p(u) > 0 when u > 0 and lin ^ w ) = оо). The sequence will be denoted

U->OQ

b y x — {£„}, у = {$„}, в = (0), xn = (tlt t2, tn, 0, 0, ...) etc. We adopt th e following notations (as in [ 2 ]):

OO

~ ^ , ^'mv(Pv{\%\)}

v = l

T ° = { x : <r*(a?)->0 as m ->00 and super^(<B)< 00},

m

Т*ф = {x: there exists a

such th a t

Т ф = {x: kceT0 for all A > 0}.

The sets T% and Т ф are linear spaces as it is easy to see. The functiona sup o*(x) for xeT°,

m

00

for xeT% — T°t

е*ф{х) =

can be defined on Т ф. This functional о*ф is satisfying the conditions:

A l.

%(

) = 0 if and only if x = 0.

A2. й (-а > ) = q I ( ho ).

A3. Q^ax + by) < £*ф{я) -j- Q%{y), where a, 0 and a + b = 1 .

B l. Q%{kc) 0 as A — >0.

2 J. A n t o n i

We shall be interested in spaces Т ф. On Т ф we can define a F-norm generated by modular дф = дФ/тф as follows:

||

| 1

= i n f > 0: q (Û

< s; •

If all functions <pv are s-convex (0 < s < 1 ), then a norm can be defined apcording to the following formula:

INI® = inf je > 0 : вФ\^1 <

The relation between the modular and the norm convergence is the following: ||ж и||ф'— >0 if and only if дФ(лхп) -> 0 for all Я > 0.

T

1 . Т ф is a complete linear space with respect to the F-norm || • ||ф.

P ro o f. Let be a Cauchy sequence, i.e. \\xp — <юй\\ф -*»0 as_p, q ^o o . The above is equivalent to the condition @ф(Л(хр — xQ)) -*0 for all Я > 0.

I t follows th at for a given &> 0 there is a positive integer n 0 such th a t

OO

SUP Z amv<PA^\tf~ta v\) < Ф for p, q ^ n0. The condition Q<p(h(xp — О

m v= l

implies th a t the sequence {l„} converges for all v and therefore there exists a sequence x — {tv}, where % — lim<“. Let m, n be positive integers.

n->co

Then for all m, n and for p , q > ?i0 the formula П

) < * / 2

is valid. From the last inequality we get

П П

lim £ a mv<pv(X\t?-t?\) == ] ? a mv<pv(X \t? -tv\) ^ s { 2 .

q->oo j r=i

00

As the last formula is true for all n, it follows th a t £ amv<pv{h \tf — $,[)

OO t > = l

< e/2 for all m. From this we conclude th a t sup ^ amv(pv(X\tf — tv\) < s.

m v= l

We proved also th a t the sequence {xn} converges to æ in norm. We are going to show th a t а>еТф:

OO OO

£ amv<Pv(*\tv\) < £ amvcpv{h 1C —^1 +Я1С1)

v = l V —1

OO

2 Я |С -У ) + ^(2Д|С1))-

v = l

As {t™}eT0, £ a mv(pv(2X |C!) ->0 as m -> oo and for all X > 0. Since

V = l 00

the sequence {xn} converges to x in norm, we have JT 1 am„<j5,,(2AjC —<„ 1 ) ->0

and х е Т ф. *’=1

R e m a rk . In the proof we have taken into account th at if {xn} is a Cauchy sequence, then the sequence {Ç} is a Cauchy sequence, too.

This statement is a consequence of regularity of the m atrix A and of the supposition about the columns of the matrix.

T

2. Let х е Т ф. Then j|& — хп\\ф-> 0.

P ro o f. Let x = {£„}, xn = (txi t2, ..., tn, 0, 0, ...). Since х е Т ф, a^{Xx) -> 0 as m ->

i.e. for a given

> 0 there is a positive integer m 0 such

OO

th at o*{xls)< e /2 for m > m 0. Since the series ^ a mv%{\tv\Je) converges, there is a positive integer v0 such th a t v==1

Then we get

for m = 1 , 2 , . . . , m0.

9 ф

x — xn e

< ^sup V (—) + ^sup V am vç>v (—)< e

' £ / m>mo ^ \ e / for all n > v0 and the proof is complete.

C

. The space Т ф is separable with respect to the norm ||*||ф.

The proof follows immediately from the inclusion T f с Т ф for any sequence Ф and Theorem 2.

Papers [1] and [ 2 ] show th a t Т фп Т ь = T v n T b if sequences Ф and W are stationary. In case of arbitrary sequences this equality does not hold as we can see from the following example.

E

. Let us put <pn(u) — un and y>n(u) = ulln. The sequence x = {1 /п }еТ ф, because

lim

m -*oo

= о

for all X > 0 ((amn) is a regular matrix). The sequence {1 jn }4 T v , because lim

m->oo

у ^{X— 1 '}

v=l

4 J. A n t o n i

This example shows th a t T 0 cannot be a subset of Т ф for any sequence Ф of ^-functions.

Now we are going to give a necessary and sufficient condition to have T 0n T b m T wn T b. The theorem containing th a t condition will be preceded by Lemma 1 To express Lemma 1, for every subset В of the set Ж of all positive integers we define

OO , 0 0

ЪЛ{В) = limsup ] ? a mvxB(v), ô A(B) = lim J ? amvxB(v)

m ->oo v==i m -+ co Vss.j

if exists and where %B is a characteristic function of th e set B.

L

1 . Let x = {%}, tv > 0 be a bounded sequence. Then lim £ amvtv

— 0 i f and only i f for every e > 0 , $A ({v: tv^ e}) = 0. w^°° V=1 P ro o f. Assume th a t for every e > 0 , SA ({v: tv^ e}) = 0 . Let us write B e = {v : tv^ e}. Let tv^ K (v = 1, 2, ...). Then we have

OO

limsup amvtv < limsup J ? u,w^ + limsup ^ amvtv

m —*oo V=1 m->oo veBe m -+ oo Vej v - S s

< КдА (Ве) + едА( Ж - В Е) = s .

OO

Since s > 0 is arbitrary, lim fff amvtv = 0 .

m—>oo v = i

Now we are going to prove the inverse implication. Suppose th a t there exists a number e > 0 such th a t 3A (Be) > 0 for some sequence {£„}.

Then we have

limsup amvtv > limsup amvtv > eôA{Be) > 0

rn-уоо у==1 m -* со veBe

and thus lim ) fa mvtv Ф 0 .

m-> oo v = l

oo oo oo oo

Let X =

/7

0 ,

and Z =

П < ~ к , К ) .

n = l n —l K = 1 n —1

A basis of neighbourhoods of point zero is to be introduced in the spaces X and Y. The basis depends on the m atrix A and consists of all sets of the form:

0 eB = {(h, h , \tp\ < e for veB }, 0 eB = {(^, t2, ...) € Y : tr <i s for v € B f

for all e > 0 and for all В с Ж such th at ôA (B) = 1. I t is easy to show th a t the above mentioned system forms a basis of neighbourhoods of zero. The function Ф: A->Y can be associated to every sequence Ф — {y>n}

of «^-functions in such a way th a t for every projection nn we have

тгпоФ = <pnQ7in, i.e. Ф((^, t2, . . . ) ) = (<Pi(\h\), <P*{\h\), •• •)•

The following characterization of the space Т ф can be given by means of the function Ф. The sequence {Q еТф if and only if the sequence {' Ч^!}€П Thi s assertion can be easily proved. According to

e>0

В

Lemma 1 the statement {tv} eТ ф is equivalent to the statement bA({v:

(pv{X\tv\) < e}) = 1 for all À > 0 and s > 0. The last relation gives {A|(,I}«U®_1(Oæ ) = {ft}: VÀ l<,l)<« for ve B)

В

for every s > 0 and all Я > 0 .

T

3. Let {(pn} and {ipn} be two sequences of <p-functions. Then T wn T b c= Т фп Т ь i f and only i f

n u ^ ) n ^ c п и ф -1(ОеВ) п г .

e>0

В

В

The function 4? is analogously defined by the sequence {yjn} as the function Ф by the sequence {qn}.

P ro o f. Let {tv} e T уГ\Тъ. Then

c= n U ф~1{оеВ) ^

e>0

В

в

for all Я > 0 and therefore {tr}eT 0 n T b.

The inverse implication can be proved by an equivalent relation.

The inclusion T ^ n T b с Т фс\Тъ is equivalent to

Уф У У дл({”- V vU ftlX *}) = 1 ~ à A({v. % w t,)< e } ) = 1 .

e>0 Л>0 '

Putting Я = 1, we have th a t y>v(\tv\) < e}) = 1 => bA[{v:

<Pr(\K\) < «}) = 1 for all {tv}eT b and every e > 0. Let 0„}еП U r _ 1 (0 £jB)n Z .

e>0

В

That means th a t for a given e > 0 there is a set В such th a t ôA(B) — 1 and %(\tv\)< e for veB . Therefore we have bA ({v: %{\t„\) < e}) = 1 . By supposition it holds ôA ({v: (pv{\tv\) < e}) = 1 . This means th a t {tv}e Г){JФ ~1{OeB) n Z and the proof of theorem is complete.

e>0 В

In the case (pn = <p and ipn = xp the condition of Theorem 2 is satis­

fied. I t can be easily verified in the following way. Let à* = inf and bf = inf {y)_ 1 (e)}. We show that

Ф~ЧО,в ) = V - \ O w > ) and Ф -'(0 < )В ) - r - \ O lS), ф-ЧО.в) = {ft}: f ( M ) < « for re B ) = {ft}: \t,\ < S' for r« B ).

r ' { 0 ^ ) = {ft}: V (ftl)< V(%) t o = {ft}: \t„\ < »1 for v * B ).

Similarly we can obtain

<t> = { M : 14 < 4 ' for re B], r - ' ( 0 . B) = {{*.}: 141 < 4 * for veB }.

Since d* and ôJ converge to zero as s -> 0 , the equation

n U *-l(Ow)nz = nu r n o ^ n z

e>0

В

В

is true. Thus, we have obtained the equality Т фслТъ — T Pr\T b.

Specifying the sequence of ^-functions as well as the matrix A we can obtain the more convenient conditions. For example, if we choose as (pn = uPn and A as the matrix of Cesàro method, then the theorems are true. ( I A = XVn.)

T

4. Lei

p

and {qn} be two convergent sequences of real number such that p n ^ a > 0, qn ^ a > 0 (n — 1 , 2, ...). Then Tp^c\Tb = T Qnn T b.

The proof is easy and we omit it.

E e m a rk . If p n ^ p (pn > p), then TPn => Tp {TPn c Tp). We can verify this assertion in the following way. Let {tv} eTp. Let us put for Я > 0, t’v = tv if À\tv\ < 1 and t' = 0 for other v and t" — tv — t'v for all v.

Then {t’9}eTPn because TPn<~\Tb = Tp r\T b. Since

then {t'v'}eTPn, and thus {tv} e TPn.

T

5. Let p n ^ a > 0, p n ->p. Then TPn = Tp if and only i f

\pn - p \ lnw = 0 ( 1 ).

Let us put p n = p + en. To prove the theorem, we will distinguish the following cases:

1 ° еп > 0 for infinite many n and en < 0 for finite many n.

2° sn > 0 for finite many n and sn < 0 for infinite many n.

3° en > 0 and en < 0 for infinite many n.

4°

> 0 and en < 0 only for finite number of n.

I t is obvious th a t in case 4° we have Tp = Tp . We are going to give the complete proof only in case 1 °; the proofs in other cases are similar.

I t is obvious th a t T„ <= Tn. We wish to show th a t T n a T n . Assume th a t

— 0(1). Let К be so chosen th a t en ln w < K . Let {tv}eTp . Since {tv}eTp , then the necessary condition for (0,1) summability, i.e.

m CO

(M tj)1

n ⁰ is fulfilled. I t follows th a t {X\tn\)p < n for sufficiently large n or psn Inn < sn Inn <

Let en > 0 for n ^ n 0. Given s > 0, there exists a positive integer m 0 > n 0 such th a t

1

m Y ^{w k}

m n.(j+l

«о ЛЛ

2 (K + l) p

<

for all m > m0. Therefore we get

1 m

m

«о

I У л т ) р* - ( т ) р m I A-i

V =1

< —-I Y W m I AmJ _*>=1

* ' - ( m r

< «.

+ — I У ( m i ) * - (m n *

«o+i

+ — \ Y w t,\)piW t,\Yn- i )

m ! A j '

«■ 0 + ¹

Thus we have proved th a t {tn}eTPn.

Next we prove the inverse implication. Assume th a t limsup |sjln w

n

= oo. The proof will be finshed if we show th a t there exists a sequence {tn} eTp such th a t {tn}4Tp . One can choose a sequence of integers which satisfies the conditions:

(a) limeftfclnw.fc =

Jc-> oo

(b) nk > (пк_г)Рпь,р, (c) lim inf--- — b > Пу, 1 .

k~><x> W 'k—1

Then we can choose a non-decreasing positive function defined on the set of all positive integers such th a t

(а) П т / ( 7 г ) = oo,

А-» oo

Ф) /(&) < — ln% for all nk where a = supp n

(t

8 J. A n t o n i

We put f(k ) = — Innk for all h if the last is inequality satisfied nk

CL

Ъу putting — ln n fc for ffk ) m it. Let us put tv = 0 for v Ф nk and

Qj

lk - 1

, Пк П,

*“* = U W ~ / ( * —i)

1/p

The sequence {tv}eTp since for nk ^ . m < nk+1 we have

m m J (k )

But the sequence {\tv\Pv} does not satisfy the necessary condition for ( 0 , 1 ) summabiliby, because there holds:

\K\Pn ^ / nk

lim su p ---> limsup — ---

n-* oo Г1- k-*oо Uk

п Ъ 1а

— limsup ih n. ₁

fc-J-00

ГЬк \f{h) f( k 1 )

1+ en klp

= limsup

f c - > OO

J \ h) > 1 .

Therefore {tv}4Tp and the proof is finished.

B ern a rk . In the case when p n oo, qn ->

as n

Theorem

is not valid.

The following theorem is a generalization of Theorem 2.4 of [ 2 ].

T

6. I f Т ф с T w and x {cT 0for all i, then \\х^\\т -> 0 i f ||а?г-||ф -> 0.

We get the theorem from the closed graph theorem and from the coordinate convergence which is a consequence of the convergence in norm.

Next we investigate spaces of strongly summable sequences.

D

. The sequence x = {tv} will be called Ф-strongly summable to number t — t(x) if {x — t}eT 0 . We shall denote the set of all Ф-strongly summable sequences by Т(Ф).

D

. The sequence Ф = {<pn} of ^-functions is called lower - regular at the point zero if there exists a ^-function xp and a fixed neigh­

bourhood U of the point zero such th a t xp(u) < inî<pn(u) for all ue U.

n

T

7. Let Ф

pn be lower-regular and equiconiinuous at the point zero. Then Т(Ф) provided with F -norm ||*||ф is a complete, normed linear space, and the coordinates are continuous functionals on Т(Ф).

First we show th a t the modular q 0 satisfies condition B1 for every

хеТ(Ф ). Suppose хеТ(Ф ), x = {tP}. Then there is a pumber t such th a t

{tv — t}e T 0 . We have

CO o o o o

X amv9v(M%\)< ]?^< р Л Ы \К ~ Ц ) + £ amv(pv(2X\t\)

V — l y = l J>=1

for all m, and thus

oo

9 ф {Л\%)< ( 2 Я

— *)) -h snp Y amv<pv{2Â\t\).

m V = 1

Since {x — t)e T 0 , Q0(À{æ — t)) ->0 as Я ^ О , it follows from the equi- continuity of the sequence Ф a t 0 th a t to a given s > 0 there is a ô > 0 such th a t <pn(u) < s/2M for u < ô and for all n (M is such a constant th a t JT \amv\ < M for m = 1 ,2 , 3, ...)• If 2X\t\ < ô, then sup ]?amv<pv(22.\t\)

v = l oo m v = l

< e. Thus we get sup ^ a mvcp(2X\t\) ->0 as Я ->0. Thus we proved th at q 0.

m v = l

satisfies condition B1 for all хеТ(Ф). I t remains to show th a t Т(Ф) is complete. Assuming oon = {%}еТ(Ф) and дФ[Х{оор — xa)) -» 0 as p, q -> oo,.

for every Я > 0 , we get tn -> t as n -> oo for every v. We show th a t {æn}

CO

converges to хеТ{Ф) in norm. Given e > 0, then sup amv<pv{X\t? —

m v = i

< e/2 for sufficiently large p and q. Therefore we have

OO

amv<pv{Â\t? -t? \) < e/ 2 , V =1

r

amv(pv(X\t?-t?\) < e /2

v — 1

for all positive integers m and r. Using the last inequality we get

r r

lim ] ? a mv<pv{À\t?-t«I) = ^ a mv<pv( À \t f - t v\)< e/2.

Qr-*OOv — i i>= 1

But the last inequality is true for all m and r, therefore we have^

OO

sup £ amv(pv(A\tf — tv\) < e, i.e. the sequence {,xn} converges to oo in norm.

m v = i

Take e > 0 and let tn = t(oon). Let щ Le such ^-function th a t щ (и)

< inf<pn{u) for every ue U, where Ü = < 0 , a); putting y{u) = y ^ u ) for П

ue U and гр(и) = lim щ(и) for и > a we obtain a function which satisfies.

u-*a~ ц->а~

yi(u) < <pn(n) for all и 3 s 0 and for all n. Therefore we have

10 J. A n t o n i

Multiplying by amv and summing we have

ww oo

I tP ~ t q\ X amv(<Pv { \t? - tp\)+<Pp(№-%\)) + ^ amv<Pv ( W ~ tq\) • V =1

Since

V =1

oo

lim ^ a mvy { l\tp - t q\) = v (l\tp - t q\)

m— >oo v—i

oo

lim ^ am(pv(\tf ~ t q\) = 0 ,

7П-+00 v=i

OO

lim £ amv<Pv( Itq - tq\) = 0 , J V =1

v= 1

^nd, the sequence {#„} is a Cauchy sequence, i.e. sup £ amv<pv{\tf —19\) < s

m v = i

io r sufficiently large p and q, thus we have yj(^ \ip — t9\) < e for sufficiently large p, q. Hence {tn} is a Cauchy sequence. Let tn ->t and X > 0. Then

we get

OO oo

У < У « т,9>ЛЗЛ |е-П ) +

OO 00

V = 1 V = 1

oo oo

< sup У ^mvtyv (ЗА\tv ^|)H“ ^mvtyv (ЗЯ 1 1

' v=l

oo

у »«,?>.( ЗЯ 1 С - П ) . V = 1

oo

Since {a?n} converges to я?, then sup £ a mv(pv(3X \iv — tf\) converges to :zero as p->oo. Suppose m v==1

CO

дф(ЗХ(х — œp)) = sup J^<V9?v(3A|t,,— if|) < e/3

for p > p i. For a given e > 0 there exists a number ô > 0 such th a t q>v(u)

< e/3 for all u < ô and all v. Assume 3X\tp — t\ < 6 for p ^ p z. Then we have

OO

lim ^ amv(pv(X\tv — t\) < e/3 + e/3 + 0 < e.

m -> oo j,_.2

Hence the last inequality is true for arbitrary e > 0 , thus we have

4>еТ(Ф).

T

8. The generalized limit t(œ) is defined uniquely and it is a modular continuous functional on Т(Ф).

P ro o f. Assume th a t there exists a sequence sc = {%}еТ(Ф) for which t(sc) = rx, t(oc) = r2 and rx ф г г. Then we have

< 9VU \ К - Гл\)+%{МК-Гь\).

Multiplying by amv and summing we get

CO

lim ^mv У*

m->oo

v=l

X

~2

\r1- r 2\ < lim

oo

CO

^am v< PvW v- r i|) +

j>=1

+ l i m V

amv<pv{ l \ t v — r 2\)

ПГП—i/Y»

Thus rx = r2, and this contradiction proves the assertion.

Now suppose сспеТ(Ф), осеТ(Ф), and, дФ[Х(ссп — ж)) -> 0 f o r a fixed A > 0 . We shall show t(scn) -+t(sc). Let q[X{œn — sc)) < e for n ^ n 0. Then we have

Multiplying by amv and summing we get

(

— \tn — t\ К lim ] ? а тР(срфА\Р-%\)+<рг{Х —$|)) +

O J m->oQ

00

+ sup y a mv(pv{ X \t^ -tv\).

m „ = 1

Therefore it holds y)(^X\tn — t\) < e for n ^ n Q and consequently t{a>n) — > t(sc).

References

£1] J. M usielak and W. O rlicz, On modular spaces of strongly summable sequences, Studia Math. 22 (1962), p. 127-146.

£2] A. W aszak, On spaces of strongly summable sequences with an Orlicz metric, Prace

Mat. 11 (1968), p. 229-246.