Vector-valued sequence space BM C ( X ) and its properties

Vector-valued sequence space BM C ( X ) and its properties

Qing-Ying Bu

Abstract. In this paper, a vector topology is introduced in the vector-valued sequence spaceBMC(X) and convergence of sequences and sequentially compact sets inBMC(X) are characterized.

Keywords: vector-valued sequence space, topology, series, compact sets Classification: 46A05, 40A05

1. Introduction

When A. Pietsch [4] gave characterizations for nuclearity of locally convex spaces in terms of vector-valued sequence spaces, he introduced a vector-valued sequence space ℓ₁(X) with values in a locally convex space X. And when Li Ronglu and Bu Qing-Ying [2] gave characterizations for a locally convex space which contains no copy of c₀, they introduced a vector-valued sequence space BMC(X) with values in a locally convex space X. In fact, ℓ₁(X) = BMC(X), the space consisting of bounded multiplier convergent series inX. From [4], [2] it is obviously seen that the spaceBMC(X) plays an important role in characterizing the structure of spaces in locally convex space theory. It will be seen in [1] that the spaceBMC(X) also plays an important role in establishing Orlicz-Pettis type theorem for compact operators on locally convex spaces.

In Section 2 of this paper, we introduce a vector topology in the spaceBMC(X) with values in a topological vector spaceX and characterize convergence of se- quences in BMC(X) and completeness of BMC(X). In Section 3 we consider the space BMC(X) with values in a locally convex space X and characterize sequentially compact subsets ofBMC(X) in different ways.

2. Convergence of sequences inBMC(X)

In this section, LetX be a separated topological vector space andU_X denote a local base of closed balanced neighbourhoods of 0 inX (see [5]). For a Banach spaceE, letB(E) denote its closed unit ball. Let

BMC(X) =n

x={xi} ∈X^N: series X

i

tixi

converges for each {t_i} ∈ℓ∞

o.

ThenBMC(X) is a sequence space with values in X. For a subsetA ofX, let Ae=n

x∈BMC(X) :X

i≥1

tixi∈A for each {ti} ∈B(ℓ∞)o

and Ue_X =n

Ue :U ∈U_Xo .

Proposition 2.1. There is a unique vector topology forBMC(X)for whichUe_X is a local base of neighbourhoods of 0. This vector topology will be denoted byτ.

Proof: By Corollary 3 of [2], for each {xi} ∈ BMC(X) the set {P

i≥1tixi : {t_i} ∈B(ℓ∞)} is compact set inX and hence, is bounded. So it follows thatUe absorbs eachxinBMC(X) for eachU ∈U_X. In addition, it is easy to see thatUe is balanced for eachU ∈U_X. And forU, V ∈U_X such thatU+U ⊂V it is easy to prove that Ue+Ue ⊂Ve. Thus we have proved Ue_X is an additive filterbase of balanced absorbing subsets ofBMC(X). Now, the proof follows from Theorem 5

of [5, p. 45].

For a net{x^α}in BMC(X), it is easy to see that

(1) τ−lim

α x^α= 0⇐⇒lim

α

X

i≥1

tix^α_i = 0 uniformly for all{ti} ∈B(ℓ∞). Let

P_k:BMC(X)−→X, P_k(x) =x_k;

I_k:X−→BMC(X), I_k(x) = (0, . . . ,0,^(k)x ,0,0, . . .).

ThenP_k andI_kare continuous linear maps,k= 1,2, . . .. Lemma 2.2 ([3]). Letxij ∈X fori, j∈N. Suppose

(I) lim_ix_ij =x_j exists for eachj∈Nand

(II) for each increasing sequence {m_j} of N there is a subsequence {n_j} of {m_j} such that{P

j≥1x_inj}^∞_i=1 is Cauchy.

Thenlimixii= 0.

Theorem 2.3. For x⁽ⁿ⁾, x ∈ BMC(X), n = 1,2, . . ., the following statements are equivalent:

(i) τ-limnx⁽ⁿ⁾=x.

(ii) limnP

i≥1t_ix⁽ⁿ⁾_i =P

i≥1t_ix_i for each{t_i} ∈ℓ∞.

(iii) limnx⁽ⁿ⁾_i =x_i for i∈ N. And for each{t_i} ∈ℓ∞, lim_kP

i>kt_ix⁽ⁿ⁾_i = 0 uniformly for alln∈N.

(iv) limnx⁽ⁿ⁾_i = xi for i ∈ N. And lim_kP

i>ktix⁽ⁿ⁾_i = 0 uniformly for all n∈Nand all{ti} ∈B(ℓ∞).

Proof: (i)⇒ (iv). By (i), limnx⁽ⁿ⁾_i =x_i obviously for i∈N. LetU, V ∈ U_X such thatV +V ⊂U. By (i), there is n₀∈Nsuch that forn≥n₀,

X

i≥1

t_i(x⁽ⁿ⁾_i −x_i)∈V, {t_i} ∈B(ℓ∞).

By Example 1 of [2], there isk₀∈Nsuch that fork≥k₀ andn= 1,2, . . . , n₀,

(2) X

i>k

t_ix⁽ⁿ⁾_i ∈U, X

i>k

t_ix_i∈V, {t_i} ∈B(ℓ∞).

So fork≥k₀ and n > n₀,

(3) X

i>k

tix⁽ⁿ⁾_i =X

i>k

ti(x⁽ⁿ⁾_i −xi) +X

i>k

tixi∈V +V ⊂U, {ti} ∈B(ℓ∞).

Thus (iv) follows from (2) and (3).

(iv)⇒(iii). Obviously.

(iii) ⇒(ii). Let {t_i} ∈ ℓ∞, U, V ∈ U_X such that V +V +V ⊂U. By (iii), there isk₀∈Nsuch that

X

i>k0

t_ix⁽ⁿ⁾_i ∈V, X

i>k0

t_ix_i∈V, n= 1,2, . . . and there isn₀∈Nsuch that forn > n₀,

k0

X

i=1

t_i(x⁽ⁿ⁾_i −x_i)∈V.

So forn > n₀, X

i≥1

t_i(x⁽ⁿ⁾_i −x_i) =

k0

X

i=1

t_i(x⁽ⁿ⁾_i −x_i) + X

i>k0

t_ix⁽ⁿ⁾_i −X

i>k0

t_ix_i ∈V +V +V ⊂U.

(ii) follows.

(ii)⇒(i). By (ii), limnx⁽ⁿ⁾_i =xi obviously fori∈N. Ifτ- limnx⁽ⁿ⁾6=x, then there would existU ∈UX, an increasing subsequence{n_k}and{t^(k)_i } ∈B(ℓ∞), k= 1,2, . . . such that

X

i≥1

t^(k)_i (x⁽ⁿ_i ^k)−xi)∈/U, k= 1,2, . . . .

For convenience, we can suppose that X

i≥1

t⁽ⁿ⁾_i (x⁽ⁿ⁾_i −xi)∈/ U, n= 1,2, . . . .

LetV, W ∈U_X such thatV +V ⊂W and W +W ⊂U. Pick m₁ ∈Nsuch thatP

i>m1t⁽¹⁾_i (x⁽¹⁾_i −xi)∈V. Then

m1

X

i=1

t⁽¹⁾_i (x⁽¹⁾_i −x_i)∈/V.

Setn1= 1. Since limnx⁽ⁿ⁾_i =xifori∈N, there isn2∈Nwithn2> n1such that Pm1

i=1s_i(x⁽ⁿ_i ²⁾−x_i)∈W for all{s_i} ∈B(ℓ∞). It follows thatPm1

i=1t⁽ⁿ_i ²⁾(x⁽ⁿ_i ²⁾− x_i)∈ W. So P

i>m1t⁽ⁿ_i ²⁾(x⁽ⁿ_i ²⁾−x_i)∈/ W. Pick m₂ ∈N with m₂ > m₁ such thatP

i>m2t⁽ⁿ_i ²⁾(x⁽ⁿ_i ²⁾−x_i)∈V. Then

m2

X

i=m1+1

t⁽ⁿ_i ²⁾(x⁽ⁿ_i ²⁾−x_i)∈/V.

Proceeding in this manner we produce increasing sequences{n_k}and{m_k}such that

(4)

mXk+1

i=mk+1

t⁽ⁿ_i ^k+1)(x⁽ⁿ_i ^k+1)−xi)∈/V, k= 0,1,2, . . . , here setm₀= 0. Let

y_kj =

mj+1

X

i=mj+1

t⁽ⁿ_i ^j+1)(x⁽ⁿ_i ^k+1)−x_i).

Then lim_ky_kj = 0 forj∈N. Sett_i=t⁽ⁿ_i ^j+1) form_j < i≤m_j+1,j = 0,1,2, . . ., andti= 0 elsewhere. Then{ti} ∈ℓ∞andP

j≥0y_kj=P

i≥1ti(x⁽ⁿ_i ^k+1)−xi). By (ii), lim_kP

j≥0y_kj = 0. So it follows from Lemma 2.2 that lim_ky_kk = 0. This contradicts (4) and (i) follows.

The proof of Theorem 2.3 is complete.

Proposition 2.4. BMC(X)is complete(or sequentially complete)space if and only ifX is complete(or sequentially complete)space.

Proof: IfBMC(X) is complete space, then it is easy to prove thatXis complete.

Conversely, ifX is complete space, we will prove thatBMC(X) is complete space.

Let{x^α}be Cauchy net inBMC(X) andU, V ∈U_X such thatV+V+V ⊂U. Then for Ve ∈ Ue_X there is α₀ such that for α, β ≥ α₀, x^α−x^β ∈ Ve, i.e. for α, β≥α₀,

(5) X

i≥1

t_i(x^α_i −x^β_i)∈V, {t_i} ∈B(ℓ∞).

By the continuity ofP_i,{x^α_i}is Cauchy net inX and hence, there isx_i∈X such that

(6) lim

α x^α_i =x_i, i= 1,2, . . . . From (5) it follows that forα, β≥α₀ and eachn∈N,

Xn i=1

ti(x^α_i −x^β_i)∈V, {ti} ∈B(ℓ∞).

So by (6) forα≥α₀ andn∈N, Xn

i=1

ti(x^α_i −xi)∈V, {ti} ∈B(ℓ∞).

Because of Example 1 of [2], there isn₀∈Nsuch that forn > n₀, X

i>n

tix^α_i⁰ ∈V, {ti} ∈B(ℓ∞).

Thus forn > n₀ andα≥α₀, Xn

i=1

t_ix_i−X

i≥1

t_ix^α_i = Xn

i=1

t_i(x_i−x^α_i)−X

i>n

t_i(x^α_i −x^α_i⁰)−

−X

i>n

t_ix^α_i⁰ ∈V +V +V ⊂U, {t_i} ∈B(ℓ∞).

It follows that the seriesP

itixi converges for each {ti} ∈ ℓ∞, i.e. x= {xi} ∈ BMC(X) and forα > α₀,

X

i≥1

t_ix_i−X

i≥1

t_ix^α_i ∈U, {t_i} ∈B(ℓ∞).

Soτ- limαx^α=xand we have proved that BMC(X) is complete. The proof is

complete.

3. Compact sets inBMC(X)

In this section, letX be a locally convex space andX^′ its dual space. Then (X, X^′) forms a dual pair. LetU_X denote a local base of barrelled neighbourhoods of 0 inX. The gauge ofU ∈U_X will be denoted byp_U and the polar ofU will be denoted byU⁰ (see [5]). It is easy to see that

(7) pU(x) = sup{|f(x)|:f ∈U⁰}, x∈X.

For eachU ∈U_X and eachx={x_i} ∈BMC(X), let

(8) ε_U(x) = supn

p_U X

i≥1

t_ix_i

:{t_i} ∈B(ℓ∞)o .

ThenεU(·) is a seminorm onBMC(X) and the topology generated by the family of seminorms{εU(·) :U ∈UX}onBMC(X) is just the original topologyτ.

Proposition 3.1. For eachU ∈U_X and eachx∈BMC(X),

(9) ε_U(x) = supn X

i≥1

|f(x_i)|:f ∈U⁰o .

The proof follows from (7) and (8).

Fort={t_i} ∈ℓ∞, let

ϕt:BMC(X)−→X, ϕt(x) =X

i≥1

tixi.

Then for eachU ∈U_X,p_U(ϕt(x))≤ε_U(x). Soϕtis a continuous linear map.

By Example 1 of [2], it is known that each{xi} ∈BMC(X) has the following property:

(∗) τ- lim

n

X

i>n

Ii(xi) = 0.

In order to consider a subset ofBMC(X), we give

Definition 3.2. A subset A of BMC(X) is called uniformly convergent if τ- limnP

i>nIi(xi) = 0 uniformly for all{xi} ∈A;Ais calledσ(X, X^′)-uniformly convergent if for eachf ∈X^′, limnP

i>n|f(xi)|= 0 uniformly for all{xi} ∈A.

Theorem 3.3. Let X be a sequentially complete space and A a subset of BMC(X). ThenAis relatively sequentially compact if and only if

(a) Ais uniformly convergent and

(b) for eachi∈N,P_i(A)is relatively sequentially compact subset ofX.

Proof: IfA is relatively sequentially compact, then (b) holds obviously. Next we will prove that (a) holds.

Suppose that (a) does not hold. Then there isU ∈U_X such that limn supn

ε_U X

i>n

Ii(xi)

:x={xi} ∈Ao 6= 0,

i.e.

limn supn

p_U X

i>n

t_ix_i

:{t_i} ∈B(ℓ∞), x={x_i} ∈Ao 6= 0.

So there are ε₀ > 0, increasing subsequence {n_k} of N, {t^(k)_i } ∈ B(ℓ∞) and x^(k)∈Asuch that

(10) p_U X

i>n_k

t^(k)_i x^(k)_i

≥ε₀, k= 1,2, . . . .

SinceA is relatively sequentially compact, there are a subsequence{x^(kj)}^∞₁ of {x^(k)}^∞₁ and x∈BMC(X) such thatτ- limjx^(kj)=x. By Theorem 2.3,

limm supn

p_U X

i>m

t_ix^(k_i ^j)

:{t_i} ∈B(ℓ∞), j∈No

= 0.

So there isn_kj such that

p_U X

i>n_kj

t^(k_i ^j)x^(k_i ^j)

< ε₀.

This contradicts (10). Thus we have proved that (a) holds.

Conversely, if the conditions (a) and (b) hold, we will prove thatAis relatively sequentially compact. Let {x⁽ⁿ⁾}^∞₁ ⊂ A. By (b), using the diagonal method we can find a subsequence {n_k} of N such that lim_kx⁽ⁿ_i ^k) exists fori ∈N. For convenience, we can suppose thatn_k=k, i.e.

(11) lim

n x⁽ⁿ⁾_i exists, i= 1,2, . . . . By (a) for eachU ∈U_X andε >0, there isk₀∈Nsuch that

εU

X

i>k0

Ii(xi)

< ε/4 for {xi} ∈A.

And furthermore, by (11) there isn₀∈Nsuch that forn, m > n₀, p_U x⁽ⁿ⁾_i −x^(m)_i

< ε/2k₀, i= 1,2, . . . , k₀. Thus forn, m > n₀,

εU x⁽ⁿ⁾−x^(m)

≤

k0

X

i=1

pU x⁽ⁿ⁾_i −x^(m)_i +εU

X

i>k0

Ii(x⁽ⁿ⁾_i ) +ε_U X

i>k0

I_i(x^(m)_i )

< ε.

So{x⁽ⁿ⁾}^∞₁ is a Cauchy sequence of BMC(X) and hence, τ- limnx⁽ⁿ⁾ exists in BMC(X) by Proposition 2.4. Thus we have proved that A is relatively sequen- tially compact. The proof is complete.

Lemma 3.4. For each t = {t_i} ∈ ℓ∞, ϕt is c.c.t. – σ(X, X^′) continuous on eachσ(X, X^′)-uniformly convergent subset ofBMC(X), where c.c.t. denotes the coordinatewise convergence topology onBMC(X).

Proof: LetAbe anσ(X, X^′)-uniformly convergent subset ofBMC(X) and{x^α} be a net ofAsuch that limαx^α_i = 0 for i∈N. Thus forε >0 andf ∈X^′, there isn₀∈Nsuch that

X

i>n0

|f(xi)|< ε/2, for x={xi} ∈A.

And hence, there isα₀ such that forα > α₀,

|f(x^α_i)|< ε/2n0, i= 1,2, . . . , n0. So forα > α₀,

|f(ϕt(x^α))| ≤

n0

X

i=1

|f(x^α_i)|+ X

i>n0

|f(x^α_i)|< ε.

Thus we have proved thatσ(X, X^′)−limαϕt(x^α) = 0. The proof is complete.

Theorem 3.5. LetX be a sequentially complete space which contains no copy ofc₀. Then a subsetAofBMC(X)is relatively sequentially compact if and only if

(c) Aisσ(X, X^′)-uniformly convergent and

(d) for eacht∈ℓ∞,ϕt(A)is relatively sequentially compact subset ofX.

Proof: IfAis relatively sequentially compact, then by the continuity ofϕtand Theorem 3.3, the conditions (c) and (d) hold.

Conversely, if the conditions (c) and (d) hold, we will prove thatAis relatively sequentially compact. Let{x⁽ⁿ⁾}^∞₁ ⊂A. By use of the proof of Theorem 3.3, we can suppose that

(12) lim

n x⁽ⁿ⁾_i =x⁽⁰⁾_i ∈X, i= 1,2, . . . . Next we will prove thatx⁽⁰⁾={x⁽⁰⁾_i } ∈BMC(X).

Forf ∈X^′, by (c) there isk₀∈Nsuch thatP

i>k0|f(x_i)| ≤1 for eachx∈A.

Since (d) implies (b),Sk0

i=1Pi(A) is a relatively sequentially compact subset ofX and hence bounded. So there is a constantc >0 such that

|f(P_i(x))|=|f(x_i)| ≤c, x={x_i} ∈A, i= 1,2, . . . , k₀.

Thus X

i≥1

|f(x_i)| ≤k₀c+ 1, x={x_i} ∈A.

Now for a fixedm∈N, by (12) there is ann₀∈Nsuch that

|f(x⁽ⁿ_i ⁰⁾−x⁽⁰⁾_i )|<1/m, i= 1,2, . . . , m.

So _m

X

i=1

|f(x⁽⁰⁾_i )| ≤ Xm

i=1

|f(x⁽ⁿ_i ⁰⁾−x⁽⁰⁾_i )|+ Xm

i=1

|f(x⁽ⁿ_i ⁰⁾)| ≤k0c+ 2.

Sincem∈Nis arbitrary, we haveP

i≥1|f(x⁽⁰⁾_i )| ≤k0c+ 2<∞. Therefore, the series P

ix⁽⁰⁾_i is a weakly unconditionally Cauchy series in X. It follows from Theorem 4 of [2] that the seriesP

ix⁽⁰⁾_i is unconditionally convergent and hence bounded multiplier convergent. Thus we have proved thatx⁽⁰⁾∈BMC(X).

Now letD=A∪ {x⁽⁰⁾}. For eacht={ti} ∈ℓ∞, since Lemma 3.4 implies that ϕtis c.c.t. –σ(X, X^′) continuous onD, by (12) we haveσ(X, X^′)−limnϕt(x⁽ⁿ⁾) = ϕt(x⁽⁰⁾). By use of the condition (d), we have limnϕt(x⁽ⁿ⁾) = ϕt(x⁽⁰⁾), i.e.

limnP

i≥1t_ix⁽ⁿ⁾_i =P

i≥1t_ix⁽⁰⁾_i . It follows from Theorem 2.3 thatτ- limx⁽ⁿ⁾ = x⁽⁰⁾. So we have proved thatAis relatively sequentially compact. The proof is

complete.

Remark 3.6. Condition (d) in Theorem 3.5 cannot be replaced by condition (b) in Theorem 3.3. For example, letX=ℓp (1< p <∞),ei= (0, . . . ,0,⁽ⁱ⁾1,0,0, . . .) andA ={(0, . . . ,0, en,0,0, . . .)}^∞₁ . Then A⊂BMC(X) and it is easy to prove that Asatisfies the conditions (b) and (c) but does not satisfy (a) and (d), and so is not relatively sequentially compact.

Acknowledgement. The author thanks his supervisors Prof. Wu Congxin and Prof. Li Ronglu for good guidance and help.

References

[1] Bu Q.Y.,Orlicz-Pettis type theorem for compact operators, Chinese Ann. of Math.17A:1 (1996), 79–86.

[2] Li R.L., Bu Q.Y.,Locally convex spaces containing no copy ofc0, J. Math. Anal. Appl.

172:1(1993), 205–211.

[3] Li R.L., Swartz C.,Spaces for which the uniform boundedness principle holds, Studia Sci.

Math. Hungar.27(1992), 379–384.

[4] Pietsch A.,Nuclear Locally Convex Spaces, Springer-Verlag, Berlin, 1972.

[5] Wilansky A., Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, 1978.

Department of Mathematics, Harbin Institute of Technology, Harbin 150006, China

(Received October 19, 1994)