Vector valued sequence spaces generated by infinite matrices

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VECTOR-VALUED SEQUENCE SPACES GENERATED

BY INFINITE MATRICES

NANDITA RATH

(Received 24 April 2000)

Abstract.LetA=(ank)be an infinite matrix with allank≥0 andPa bounded, positive real sequence. For normed spacesEandEkthe matrixAgenerates paranormed sequence spaces such as[A, P ]_∞((Ek)), [A, P ]0((Ek)), and[A, P ](E)which generalize almost all the existing sequence spaces, such asl_∞, c0, c, lp, wp, and several others. In this paper, conditions under which these three paranormed spaces are separable, complete, andr -convex, are established.

2000 Mathematics Subject Classification. 40A05, 40C05, 40H05, 46A35, 46A45.

1. Introduction. The study of sequence spaces is generally initiated by problems in summability theory, Fourier series and power series. During the third decade of the present century, sequence spaces were studied with more insight and vision through the application of functional analysis. Rich settings for the analytic approach to the study of sequence spaces have already been provided by pioneers like Banach (see [1]), Köthe and Toeplitz (see [6,7,8]). Consequently, the study of sequence spaces is now generally regarded as a branch of linear topological spaces. But it is not hard to realize that it has more intimate relation with the summability theory and matrix transfor-mation than any other area.

One of the classic problems in the theory of sequence spaces is to transform one sequence space into another and study the behavior pattern of the transformed se-quences related to the original space. A decisive break with the classical approach is made in this paper by introducing vector-valued sequences in place of sequences of numbers. We study the sequence spaces which are generated by infinite matri-ces. These spaces are linear topological spaces, the topologies of which are induced by paranorms. The paranormed sequence spaces were introduced by Borwien (see [2,3]), Bourgin (see [4]), Simons (see [16,17]), and later, Maddox (see [9,10,11,12,

13,14]) developed it in considerable details. We study certain topological properties like separability, completeness, andr-convexity of these generalized paranormed se-quence spaces. In fact, all the corresponding results related to the sese-quence spaces c0, lp, c, l∞, andwpfollow as special cases of the theorems established here. Stated otherwise, sequence spaces are studied in this paper with a new approach and insight.

k∈Nand for the matrixA=(ank),ank≥0 for alln, k∈N. IfPis a bounded sequence, we writepk=O(1)andM=max(1,supkpk). It should be noted that all the results remain valid, if we assume thatEandEkare seminormed orp-normed spaces, while Theorem 4.10is established for seminormed spaces. The conditions under which the vector-valued sequence spaces arer-convex, when eachEkisp-normed, are yet to be explored.

The following spaces are frequently used:

[A, P ]∞Ek

=

x=xk

∈Eksup n

k

ankxkpk<∞

,

[A, P]0

Ek

=

x=xk

∈Ek

k

ankxkpk exists,∀n∈Nand →0 asn→∞

,

[A, P ](E)=

x=xk xk∈E,∀kand∃l∈Esuch that

k

ankxk−lpkexists∀n∈Nand →0 asn → ∞

.

(2.1)

These spaces are calledvector-valued sequence spaces generated by infinite matrices. In the special case, whenA=I(resp., the Cesaro matrix(C,1)),[A, P ]∞((Ek))reduces tol∞(P , Ek)(resp.,w∞(P , Ek)). The spacel(P , Ek)is obtained by considering the matrix

A=(ank), whereank=1, for allk, 1≤k≤nandank=0, for allk > n. Similarly, we have the following spaces:

[I, P ](E)=c(P , E), [I, P ]0

Ek

=co

P ,Ek

,

(C,1), P 0

Ek

=w0

P ,Ek

, (C,1), P (E)=w(P , E). (2.2)

These representations are not unique, because we can also write [B, P ]0((Ek))=

l(P , (Ek)), whereB=(bnk)is the matrix defined asbnk=1, ifk > nand bnk=0, if 1≤k≤n. Similarly, the spaces w(P , E), w0(P , E), andw∞(P , (Ek))can be repre-sented as[D, P ](E), [D, P ]0((Ek)), and[D, P ]∞((Ek)), respectively, whereD=(dnk) is the matrix defined bydnk=2−(n−1), if 2n−1≤k <2n and dnk=0, otherwise. A few decades ago, Maddox introduced [A, P ]0(resp., [A, P ], [A, P ]∞) which is a

spe-cial case of[A, P ]0((Ek))(resp.,[A, P ](E), [A, P ]∞((Ek))), whenEk=C, for allk≥1. The conditions under which the three spaces[A, P ]0, [A, P ], and[A, P ]∞are linear

paranormed can be found in [12]. These results hold without any substantial change for the vector-valued sequence spaces[A, P ]0((Ek)), [A, P ](E), and[A, P ]∞((Ek)). If

pk=O(1), then each of these three spaces are linear topological spaces, the topology being induced by a paranormgdefined by

g(x)=sup n





k

ankxkpk  

1/M

, (2.3)

3. Lemmas. The following lemmas are used in proving the theorems of this paper.

Lemma_3.1[9]. _{A linear topological space isr-convex for somer >1if and only ifX} is the only neighborhood of the origin.

Lemma_3.2[5]. _{Letx, y, λ, µbe complex numbers. Then}

|x+y|p_{≤ |x|}p_+|y|p_{, if0< p≤1,}

|λx|+|µy|p≤ |λ||x|p_+|µ||y|p_{, ifp≥1,|λ|+|µ| ≤1.} (3.1)

Lemma3.3[15]. Ifxis a complex number with0<|x| ≤1,0< p≤r, anda >1, then

|x|p<|x|r(1+aloga)+aπ, (3.2)

where1/π+r /p=1.

Lemma3.4[11]. The spacew_∞(p)is paranormed byg(x)=sup_n[_ka_nk|x_k|pk_]1/M_,

whereM=max(1,supkpk)if and only if 0<infkpk≤supkpk<∞.

Lemma_3.5. _{LetE be a nontrivial space. Then[A, P ](E)}⊆_{[A, P ]∞(E)if and only if}

α=supn

kank<∞.

Lemma_3.6. _LetB=_{(bnk)be any matrix of zeros and ones and letr be any positive} number. IfBis any column finite matrix and

B, (r ) _∞Ek⊆[B, P ]∞Ek, (3.3)

where(r )=(r , r , r , . . . ), then there exists an integeri >1such thatsupn

s(n)a πk

i < ∞, where1/πk+r /pk=1, fork∈Nands(n)= {k|bnk=1, pk< r}, for eachn∈N.

Lemma _3.7. _{The space l∞(P , (Ek)) is a linear topological space if and only if}

infkpk>0.

The analogue of Lemmas3.5,3.6, and3.7for the spaces[A, P ]_∞, [B, P ]_∞, andl_∞(P ) can be found in [9,10,12], respectively.

4. Main results. This section deals with the results established in this paper. The necessary and sufficient conditions for separability, completeness, andr-convexity of the vector-valued sequence spaces[A, P ]0((Ek)), [A, P ](E),and[A, P ]∞((Ek))are obtained in Sections4.1,4.2, and4.3, respectively.

4.1. A topological space is said to beseparable, if it has a countable dense subset. In this subsection, we obtain necessary and sufficient conditions for separability of the spaces[A, P ]0((Ek))and[A, P ](E). In general, the space[A, P ]∞((Ek))is not separable, since as a special case of this spacel_∞is not separable.

Theorem_4.1. _{Letlimn→∞ank}=_0andLk=_{supnank>0, for each fixedk}∈N_{. Then}

[A, P ]0((Ek))is separable if and only if eachEkis separable.

Proof. ₍⇐_{). Suppose that eachEkis separable. Let} Ꮾ_{be the set of all finite}

we show thatᏮhas a countable dense subset. Since eachEkis separable, we can find a countable dense subsetFk⊆Ek,for eachk∈N. LetF denote the set of all finite sequences inkFk. Clearly,F is a countable subset ofᏮ. Also, ify=(y1, y2, . . . , yr, 0,0,0, . . .)∈Ꮾ, we choosezk∈Fksuch that

yk−zkpk< , (4.1)

for each 1≤k≤r.

Letz=(z1, z2, . . . , zr,0,0,0, . . .)∈F. Since limn→∞ank=0, for each fixedk∈N,

g(y−z) M=sup n

r

k=1

ankyk−zkpk< µ, (4.2)

whereµ=supn r

k=1ankis a finite constant (depending on the sequencey). Hence, it follows that[A, P ]0((Ek))is separable.

(⇒). Conversely, letDbe a countable dense subset in[A, P ]0((Ek)). For each fixed

r∈N, letDr= {yr|y=(yk)∈D}. We show thatDris dense inEr. Letx∈Er. Define a sequencexr_by

xr k=

  

x, ifk=r ,

0, ifk≠r . (4.3)

Then, xr _{∈[A, P ]}

0((Ek)), since limn→∞anr =0. For a given > 0, we can choose

y=(yk)∈F such that

g(y−x)r=sup n

k

ankyk−xkpk 1/M

<pr_L

r 1/M

. (4.4)

Therefore,

sup n

anryr−xpr < prLr, (4.5)

which implies thatyr−x< . This completes the proof ofTheorem 4.1.

Köthe [6,7] obtained the necessary and sufficient condition for the separability of lp(E), for 1< p <∞. This can be deduced from the following corollary which is a direct consequence ofTheorem 4.1.

Corollary4.2. The spacel_p((E_k))(also,c₀(P , (E_k)), w₀(P , (E_k))is separable if and only if eachEkis separable. In particular, each oflp(E), cp(E), andwp(E)) is separable if and only ifEis separable.

Theorem_4.3. _LetL=_infkpk>0,H=_supnkank<∞_{,limn→∞ank}=_{0, for each} fixedk∈Nandᏸr=supnanr≥0, for at least oner∈N. Then,[A, P ](E)is separable if and only ifEis separable.

Proof. ₍⇐_{). LetB}=_{(xi)be a countable dense subset ofE and letF denote the}

set of all ultimately constant sequences of elements ofB, that is, all sequences of the typeyr_=(x

Lety=(yk)∈[A, P ](E)and let 0< <1. For eachk∈N, choosexki∈Bsuch that _yk−xi

k

pk_≤M_{. (4.6)}

Then, there existsl∈Esuch that

lim n→∞

k

ankylk

pk_{=0. (4.7)}

So, we can findk0∈N, for which

sup n∈N

∞

k=k0+1

ankyk−lpk≤M, (4.8)

andb∈Bsuch that

l−bL_≤M/(M−1)_{. (4.9)} Definez=(zk)by

zk=   

x_ki, if 1≤k≤k0,

b, ifk > k0.

(4.10)

Clearly,z∈F. Also, it follows from (4.6) and Minkowski’s inequality that

sup n∈N

k

ankyk−zkpk 1/M

≤sup n∈N

k=1

k0ankyk−xki pk

1/M +sup

n∈N

_∞

k=k0+1

ankyk−bpk 1/M

≤µ1/M+sup n∈N

_∞

k=k0+1

ankyk−lpk 1/M

+sup n∈N

_∞

k=k0+1

ankl−bpk

1−(1/M)

,

(4.11)

sincepk≤MandM≥1.

Now, considering the inequalities in (4.8), (4.9), and sincel≤pkwe can have

g(y−z)≤µ1/M++µ1/Msup k

pk/L_{. (4.12)}

This proves that[A, P ](E)is separable.

(⇒). Conversely, let[A, P](E)be separable with a countable dense subsetD=(yi₎ i∈N, whereyi_=(yi

k)k∈Nfor eachi∈N. Forx∈E, letxr _{denote the sequence}

x_kr=   

x, ifk=r ,

0, otherwise. (4.13)

Clearly,xr _{∈[A, P ](E). Then, as inTheorem 4.1, we can show that the set G= {y}i r|

r∈N}is dense inE. This completes the proof ofTheorem 4.3.

4.2. A paranormed space is said to becomplete, if every Cauchy sequence con-verges. This subsection deals with the completeness of the generalized sequence spaces[A, P ]∞((Ek)), [A, P ]0((Ek)), and [A, P ](E). In Theorem 4.5(i), we show that completeness of each spaceEkimplies completeness of[A, P ]0((Ek))and[A, P ]∞((Ek)), while in part (ii) (resp., (iii)), conditions for which completeness of[A, P ]∞((Ek))(resp.,

[A, P ]0((Ek))) implies completeness of eachEkare established. Finally, inTheorem 4.8, completeness of[A, P ](E)is discussed.

Theorem_4.5. _LetLk=_supnank≥_{0, for eachk}∈N_{. Then the following statements} are true:

(i) The spaces[A, P ]0((Ek))and[A, P ]∞((Ek))are complete, whenever the spaces

Ekare complete for eachk∈N.

(ii) The spacesEk,for eachk∈Nare complete, whenever[A, P ]∞((Ek))is complete andLk=supnank<∞, for eachk∈N.

(iii) The spacesEk, for eachk∈Nare complete, whenever[A, P ]0((Ek))is complete andlimn→∞ank=0.

Proof. (i) Letxi=(xi_k)_k∈N be a Cauchy sequence in[A, P ]_∞((E_k)). For a given

≥0, leti0be such that

sup n

k

ankxki−x j k

pk_≤M_{, i, j≥i}

0. (4.14)

Ifkis such thatLk≥0, then

Lkxik−x j k

p_k

≤M_{, ∀i, j > i}

0, (4.15)

which shows that(xi

k)i∈Nis a Cauchy sequence inEk, for eachk. Let

yk=   

limi→∞xik, ifLk>0,

any element ofEk, ifLk=0.

(4.16)

Then, it follows from (4.11) that

sup n

k

ankxik−yk p_k

≤M_{, ∀i > i}

0. (4.17)

Hence,y=(yk)∈[A, P ]∞((Ek))andxi→y in[A, P ]∞((Ek)), asi→ ∞. This proves the completeness of[A, P ]∞((Ek)). The completeness of[A, P ]∞((Ek))can be proved by using a similar argument.

(ii) Let[A, P ]∞((Ek))be complete andk∈Nbe fixed such that 0< Lk<∞. Ifx=

(xk)is a Cauchy sequence inEkand for eachr∈N,yr denotes the sequence whose

rth term isxr and all other terms are zero, thenyr∈[A, P ]∞((Ek)). Moreover, the sequenceyi_{whoseith term isy}

ifor eachi∈Nis a Cauchy sequence in[A, P ]∞((Ek)), because

gyi−yj=sup n

ankxi−xjpk 1/M=Lkxi−xjpk 1/M, (4.18)

which converges to zero as i, j → ∞. Since [A, P ]_∞((Ek)) is complete, there exists

z=(zk)∈[A, P ]∞((Ek))such that

This implies that

Lkxi−zkpk →0, asi→ ∞. (4.20)

Hence,xi→zk, asi→ ∞. This establishes the completeness ofEk. The proof of (iii) is similar to the proof of (ii). This completes the proof ofTheorem 4.5.

Corollary _4.6. _{Each Ek is complete if and only if any of the spaces l(P , (Ek)),}

c0(P , (Ek)),w0(P , (Ek)), andl∞(P , (Ek))is complete. In particular,c0, lp, l∞, andw0p are complete.

Corollary4.7. Letlim_n→∞a_nk=0andL_k=sup_na_nk>0, for somek∈N. Then,

Ekis complete, whenever[A, P ](E)is complete.

Proof. _{This follows from part (iii) ofTheorem 4.5upon noting that[A, P ]0(E)is}

complete, whenever[A, P ](E)is complete.

The next theorem establishes the conditions for the completeness of [A, P ](E), wheneverEis complete.

Theorem_4.8. _Letsupnkank<∞_{. Then, completeness ofEimplies that[A, P ](E)} is complete, whenever any of the following conditions hold:

(i) limn→∞kank=0, (ii) lim supn

kank>0andinfkpk=L >0.

We omit the proof ofTheorem 4.8, since the proof is exactly similar to that given by Maddox (see [12, Theorem 5]). However, the removal of the restriction infkpk>0 is possible in some special cases such asc(P , E)andw(P , E). In fact, the next theorem shows that part (ii) ofTheorem 4.8holds for these two spaces without the restriction infkpk>0. So it generalizes a result of Maddox (see [12, Theorem 6]).

Theorem_4.9. _{(i)The spacec(P , E)is complete if and only ifEis complete.}

(ii) The spacew(P , E)is complete if and only ifEis complete.

Proof. _{(i) Let(x}i_{)be a Cauchy sequence inc(P , E). So, for eachx}i_{inc(P , E), there}

existsli_{∈Esuch that x}i

k−li

pk _{→0, ask → ∞for eachi∈N. (4.21)}

Then, as inTheorem 4.8(ii), we can findx∈l∞(P , E)such thatg(xi−x)→0, asi→ ∞.

So it suffices to show that x ∈c(P , E). The case infkpk>0 can be deduced from Theorem 4.8as a special case, when the matrixA=I.

Let infkpk=0 andqk=pk/M, for eachk∈N. Choose an integeri0such that

gxi−xi0_<1

6, ∀i > i0. (4.22)

Takingi > i0andksufficiently large, we get x_ki−liqk_<1

6, x i0 k −li0

q_k

<1

6. (4.23)

So it follows that

sk=li−li0 q_k

<1

for eachk∈N. But, since infkqk=0, it follows thatsk>1/2, for infinitely manyk∈N, unlessli_=li0_{, for all sufficiently largei. This implies that(l}i_{)is ultimately a constant} sequence. Hence,

xk−li0pk/M≤xki−xkpk/M+xki−li

pk/M_+li_−liopk/M_{, (4.25)}

which converges to zero ask→ ∞, implying thatx∈c(P , E).

Conversely, letc(P , E)be complete. Letx=(xn)be a Cauchy sequence inE and let yn _{denote the sequence whose first term is x}

n and all other terms are zero. Clearly,yn_{∈c(P , E), for alln∈Nand (y}i_{)is a Cauchy sequence inc(P , E). Since}

c(P , E)is complete, there existsz=(zk)inc(P , E)such thatg(y−z)→0, asi→ ∞. This implies that

xi−z1 →0, asi→ ∞, (4.26)

and therefore,Eis complete.

(ii) Let (xi)be a Cauchy sequence in w(P , E)with li as the strong Cesaro limit ofxi_{, for eachi∈N. Then by Theorem 4.8(ii), there existsx∈w}

∞(P , E) such that

g(xi_{−x)→0, asi→ ∞. So it suffices to show thatx∈w(P , E). Choosei}

0for which

2−r 2r_≤k<2r+1

xi k−x

j k

pk_{< , ∀i, j > i}

0. (4.27)

Since for each fixedi, j > i0, there existsr0such that

2−r

2r_≤k<2r+1 _xi

k−li

pk_{< , 2}−r

2r_≤k<2r+1 _xi

k−lj

pk_{< , (4.28)}

it follows that

2−r

2r_≤k<2r+1

li−ljpk_<3M_{, (4.29)}

for eachi, j > i0and allr > r0. Also, for 3M <1/2, _li_−lj_{<1, ∀i, j > i}

0, (4.30)

which implies that

li_−ljµ_< 2r_≤k<2r+1

2−r_li_−ljpk_<3M_{, (4.31)}

for alli, j > i0, whereµ=supkpk. So(li)is a Cauchy sequence inE. SinceEis complete, there existsl∈Esuch thatli_{→l, asi→ ∞, for somel∈E. It now remains to show} thatxk→l[w(P , E)]. If we denote byNr(α)the number ofkin the interval[2r,2r+1) such thatpk< α, then we have the following two possibilities:

inf α>0

lim sup r→∞ 2

−r_N

r(α)=0 (4.32)

or

inf

α>0lim supr→∞ 2 −r_N

In the case (4.32), for given >0, there existsα0>0 such that lim supr→∞2−rNr(α0) <

/2. Therefore, 2−r_N

r(α0) < , for all sufficiently large values ofr. Now chooseiso

large that

li_−l<min1, 1/α0_{. (4.34)}

Hence, for all sufficiently larger,

2−r 2r_≤k<2r+1

l−lipk_≤2−r pk<α0

li_−lpk₊₂−r pk≥α0

li_−lpk

<2−r pk<α0

1+2−r pk≥α0

<2−rNr

α0

+ <2,

(4.35)

which implies that

2−r

2r_≤k<2r+1

l−lipk _{→0, asr → ∞. (4.36)}

Since for anyi∈N,



2−r 2r_≤k<2r+1

xk−lpk  

1/M

≤gx−xi₊ 

2−r 2r_≤k<2r+1

xi_k−lipk

 

1/M

+ 

2−r

2r_≤k<2r+1

_l−lipk

 

1/M

,

(4.37)

by choosingi, r sufficiently large, it can be shown thatxk→l[w(P , E)].

In the case (4.33), if we chooseβ=infα>0lim supr→∞2−rNr(α) >0, then there exists

r1such that 2−rNr(1) > β, forr=r1and there existsr2such that 2−rNr(1/2) > β, forr=r2and so on. In fact, this determines a sequence of integersr1< r2< r3<···,

such that

2−r_N r

₁

s

> β, (4.38)

for eachr=rsand eachs∈N. By inequalities (4.29) and (4.30), there exists an integer

t=t(β)such that

2−r

2r_≤k<2r+1

li−ljpk_<β

2, (4.39)

for sufficiently largerandli_−lj_{<1, for alli > t. Now we must havel}i_=lt_{, for all}

i > t. Because, otherwise we haveli_−lt_{>1/2, for somei > t, which implies that}

2−r 2r_≤k<2r+1

li_−ltpk_≥2−r pk<1/s

li_−ltpk

≥2−r_N r

₁

s

li_−lt1/S_{> βl}i_−lt_>β 2,

(4.40)

4.3. A subsetVof a linear topological spaceXis said to beabsolutelyr-convex(we sayr-convexfor brevity), ifλx+µy∈Vwheneverx, y∈Vandλ, µare scalars such that|λ|r_+|µ|r_{≤1. A linear topological spaceXis said to ber-convex, if the family of} allr-convex neighborhoods ofθform a neighborhood base. This section establishes the results related to ther-convexity of the spaces[A, P ]∞((Ek))and[A, P ]0((Ek)). Maddox and Roles (see [15, Theorem 4]) have already obtained necessary and sufficient conditions for ther-convexity of[A, P ]∞, which is a special case ofTheorem 4.10. Note

that byLemma 3.1, we need to consider only the case 0< r≤1, while characterizing ther-convexity of the space[A, P ]∞((Ek)).

Theorem 4.10. LetA be a column finite matrix and suppose that there exists a constantα >0such that for eachnandkwith0<supnank<∞andank>0, we have

ank≥αsupnank. If0< r≤1, then the following are equivalent: (i) The space[A, P ]_∞((Ek))isr-convex.

(ii) [µ, (r )]∞((Ek))⊆[µ, P ]∞((Ek)), whereµ=(hnk)is the matrix defined byhnk=1, if0<supnank<∞,ank>0, andhnk=0, otherwise.

(iii) There exists an integeri >1such that

sup n

s(n)

iπk_<∞_{, (4.41)}

wheres(n)= {k|ank>0, pk< r ,supnank<∞}, for eachn∈Nand1/πk+r /pk=1, for eachk∈N.

Proof. _{Since it follows immediately fromLemma 3.6that (ii) implies (iii), we only}

show (i) implies (ii) and (iii) implies (i).

(i)⇒(ii). Letx∈[µ, (r )]∞((Ek)). Then there existsv≥1 such that

sup n

k

hnkxkr≤v. (4.42)

Since[A, P ]∞((Ek))isr-convex, there exist anr-convex neighborhoodUof the origin and a real numberd >0 such thats(d)⊆U⊆s(1). Letk∈Nbe such that 0<supnank< ∞. Define the sequence(yk_)byyk_=[dM_/sup

nank]1/pk(0,0,0, . . . , xk/xk,0,0, . . .), if

xk≠0 andyk=(0,0,0, . . .), ifxk=0. Clearly,(yk)∈s(d)⊆U. Writingλk= xkv−1/r for eachk∈Nand ˜kfor any finite sum overkfor whichhmk=1,m≥1 being a fixed integer, we see that

˜

k

|λ|r=˜

k

hmkxkrv−1≤1. (4.43)

Then it follows from ther-convexity ofUthatkλkyk∈U. SinceU⊆s(1), we have

˜

k

amk|λ|pk

dM supnank

≤1, (4.44)

which implies that

˜

k

for each finite sum overkfor whichhmk=1. Now, sincev≥1,

˜

k

hmkxkpk≤vM/rα−1d−M. (4.46)

This is true for everym∈N. So it follows thatx∈[µ, P ]∞((Ek))and consequently, (ii) follows.

(iii)⇒(i). Suppose that (4.41) holds. To show ther-convexity of [A, P ]_∞((Ek)), we construct anr-convex neighborhood base atθ∈[A, P ]∞((Ek)). For eachk∈N, let

qk=maxk(r , pk)and for each 0< d <1, define

U1(d)=

x∈[A, P ]∞Ek sup n

k

ankxkpkqk/pk≤d

,

U2(d)=

x∈[A, P ]∞Ek sup n,k

ankxkpkqk/pk≤d

,

(4.47)

andU (d)=U1(d)∩U2(d). Ifx, y∈U (d)and|λ|r+|µ|r≤1, then by considering the

casesqk<1 andqk≥1 separately and applyingLemma 3.2, we obtain

λxk+µykqk≤ |λ|rxkqk+|µ|rykqk. (4.48)

Therefore, we have

sup n

k

ankλxk+µykpk q_k/p_k

≤|λ|r_+|µ|r_{d≤d, (4.49)}

which implies thatλx+µy∈U1(d). Similarly, we show thatλx+µy∈U2(d). Hence

U (d)isr-convex. Sinces(d1/M_{)⊆U (d)whenever 0< d <1, it follows thatU (d)is} a neighborhood ofθ. To prove that the set of all the U (d), for 0< d <1, form a neighborhood base atθ, it suffices to show that for each >0, there exist 0< d <1 such thatU (d)⊆s(). Lett(n)= {k∈s(n)|pk< r /2}. Since−1≤πk<0, for each

k∈t(n), it follows from (4.41) that t(n)is a finite set for eachn. Let u(n)be the number of elements int(n). Since

s(n)

iπk≥

t(n)

i−1=i−1u(n), (4.50)

for each n, it follows that µ1=supnu(n) <∞. If x ∈U (d) for some 0< d <1, observe that

pk≥r

ankxkpk=

pk≥r

ankxkpk qk/pk≤d,

pk<r /2

ankxkpk≤sup n,k

xkpk

pk<r /2

1≤d µ1.

(4.51)

Also, sinceqk=rforr /2≤pk< r, it follows fromLemma 3.3that

r /2≤pk<r

ankxkpk=

r /2≤pk<r

a1/pk

nk xk pk

≤

r /2≤pk<r

a1/pk

nk xkr

(1+TlogT )+ r /2≤pk<r

Tpik_,

for eachT >1. Ifr /2≤pk< r, thenπk<−1, so that

r /2≤pk<r

(jT )πk≤_j−1

s(n)

Tπk_{, (4.53)}

for any positive integerj. Therefore, supn

r /2≤pk<rT

π_{kcan be made arbitrarily small}

by a suitable choice ofT and by the fact thatµ1<∞. For a given >0 chooseT >1

such that

sup n

r /2≤pk<r

iπk_<

2 M

, (4.54)

and choose 0< d <1 such that

d2+µ1+TlogT

<

2 M

. (4.55)

Then,

g(x)≤ 



p_k≥r

ankxkpk+

pk<r /2

ankxkpk+

r /2≤pk<r

ankxkpk  

1/M

≤

d+µ1d+(1+TlogT ) d+

2 M1/M

< ,

(4.56)

which shows thatx∈s().

This completes the proof ofTheorem 4.10.

Unlike other properties,r-convexity of the sequence space[A, P ]_∞((Ek))does not depend on ther-convexity of the spaceEk.

Remark4.11. InTheorem 4.10we may replace the sequence space[A, P ]_∞((E_k)),

leaving the rest unchanged. The new result is still valid.

Corollary_4.12. _{The following statements are equivalent:}

(i) l(P , (Ek))isr-convex;

(ii) 0< r≤1andl((r ), (Ek))⊆l(P , (Ek));

(iii) 0< r≤1and there exists an integeri >1such that

k

iπk_<∞_{, (4.57)}

where1/pk+r /πk=1for eachkand the summation is overksuch thatpk< r.

The proof is analogous to a result in [14, Theorem 1], which is special case of

Theorem 4.10.

Corollary_4.13. _{The following statements are equivalent:}

(i) w∞(P , ((Ek)))isr-convex.

(ii) 0< r≤1, 0<infkpk, and[B, (r )]∞((Ek))⊆[B, P ]∞((Ek)), where bnk=1for 2n−1_{≤k <2}n_andb

nk=0otherwise.

(iii) 0< r≤1,infkpk>0, and there exists an integeri >1such that

sup n

s(n)

iπk_<∞_{, (4.58)}

wheres(n)= {k|2n−1_{≤k <2}n_{, p}

Proof. _{Letw∞(P , ((Ek)))ber-convex. Then byLemma 3.4, 0<infkpk}≤_supkpk<∞_.

Sincew∞(p, (Ek))is not the only neighborhood of the origin, byLemma 3.1, 0< r≤1. The rest of the proof of this corollary follows fromTheorem 4.10withA=D, where D=(dnk)is the matrix defined by

dnk=   

2n−1_{, if 2}n−1_{≤k <2}n_{for eachn,}

0, otherwise. (4.59)

Here we make use of the fact that both the matricesDand the Cesaro matrix(C,1) generate the same paranorm topology onw∞(P , (Ek))(cf. [15, page 70]).

Corollary_4.14. _{The spacel∞(P , (Ek))is1-convex if and only ifinfkpk>0.}

Proof. _{ByLemma 3.7,l∞(P , Ek)is a linear topological space if and only if infkpk>}

0. The rest of the proof can be deduced fromTheorem 4.10by puttingA=Iandi=2 in (4.41).

ByRemark 4.11, we have the following result.

Corollary_4.15. _{The sequence spacec0(P , (Ek))is1-convex.}

There are several other topological properties of the vector-valued sequence spaces [A, P ]∞((Ek)), [A, P ]0((Ek)), and[A, P ](E), which still remain to be investigated. The construction of continuous duals and Köthe-Toeplitz duals of these spaces will also be interesting, since these spaces generalize the existing sequence spaces. Needless to say, there can be many applications of these three generalized sequence spaces in the study of topological and geometric properties of all our real and complex sequences.

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Nandita Rath: Department of Mathematics and Statistics, University of Western

Australia, Nedlands, Australia