DUAL PAIRS OF SEQUENCE SPACES

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DUAL PAIRS OF SEQUENCE SPACES

JOHANN BOOS and TOIVO LEIGER (Received 31 January 2001)

Abstract.The paper aims to develop for sequence spacesEa general concept for recon- ciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE^×(× ∈ {α, β})combined with dualities(E, G),G⊂E^×, and theSAK- property (weak sectional convergence). TakingE^β:= {(y_k)∈ω:=K^N|(y_kx_k)∈cs} =: E^cs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofE^cs by replacingcsby any locally convex sequence spaceSwith sums∈S(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E, E^S)of sequence spaces and gives rise for a generalization of the solid topol- ogy and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality(E, E^β). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and gen- eralized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justi- fied by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.

2000 Mathematics Subject Classification. 46A45, 46A20, 46A30, 40A05.

1. Introduction. In summability as well as in investigations of topological sequence spacesEthe duality(E, E^β), whereE^βdenotes theβ-dual ofE, plays an essential role.

For example, if anFK-spaceEhas theSAK-property (weak sectional convergence), then the topological dualEcan be identified withE^β. Further and more deep-seated con- nections between topological properties of the dual pair(E, E^β)and theSAK-property, the continuity of matrix maps onEand the structure of domains of matrix methods have been presented, for example, in well-known inclusion theorems by Bennett and Kalton [3, Theorems 4 and 5] (see also [5,6] for generalizations).

TheSAK-property has been generalized and modified by several authors in dif- ferent directions whereby several generalizations and modifications of the notion of the β-dual has been treated: Buntinas [8] and Meyers [15] as well as further au- thors have investigated theSTK-property (weakT-sectional convergence) inK-spaces Eand the correspondingβ(T )-duals, whereTis an Sp1-matrix; Fleming and DeFranza [10,11] have dealt with theUSTK-property (unconditionally weakT-sectional conver- gence) and the correspondingα(T )-dual in case of an Sp^∗₁-matrix T. That complex of problems is also connected with theUSAK-property of sequence spaces (cf. Sem- ber [19], Sember and Raphael [18] as well as Swartz [20], Swartz and Stuart [21]), in particular properties of the duality (E, E^α) where E^α denotes the α-dual of E;

moreover, Buntinas and Tanovi´c-Miller [9] investigated the strong SAK-property of FK-spaces.

In the present paper, we define and investigate—on the base of the general notion of a sum introduced by Ruckle [17]—dual pairs(E, E^S)whereEis a sequence space,Sis aK-space on which a sum is defined in the sense of Ruckle, andE^Sis the linear space of all corresponding factor sequences. In this connection we introduce and study in Sections3and5theSK-property which corresponds with theSAK-property and the so-called quasi-matrix mapsA, respectively. In particular, we describe the continuity ofAand a natural topological structure of the domainFAofAwhereFis aK-space. By means of those results, inSection 6we formulate and prove in that general situation the mentioned inclusion theorems as well as a further theorem of Bennett-Kalton type due to Große-Erdmann [12]. The fact that all mentioned modifications of the SAK-property and of theβ-dual are special cases of theSK-property and the factor sequence spaceE^S, respectively, enables us to deduce inSection 7from the general inclusion theorems, proved in this paper, those in the listed special cases.

2. Notation and preliminaries. The terminology from the theory of locally convex spaces and summability is standard, we refer to Wilansky [23,24].

For a given dual pair(E, F )of linear spacesEandFoverK(K:=RorC)we denote byσ (E, F ),τ(E, F ), andβ(E, F )the weak topology, the Mackey topology and the strong topology, respectively. If(E, τE)is a given locally convex space, thenE^∗andEdenotes respectively, the algebraic dual ofEand the topological dual of(E, τE).

A sequence space is a (linear) subspace of the spaceω of all complex (or real) sequences x =(xk). The sequence spaceϕ is defined to be the set of all finitely nonzero sequences. Obviously, ϕ=span{e^k|k∈N}, where e^k:= (0, . . . ,0,1,0. . .) with “1” in thekth position, andϕcontains obviously for eachx∈ωits sections x^[n]:=n

k=1xke^k(n∈N).

If a sequence spaceE carries a locally convex topology such that the coordinate functionalsπn (n∈N) defined byπn(x)=xn are continuous, thenE is called a K-space. For everyK-spaceE the spaceϕis aσ (E, E)-dense subspace of E where ϕis identified with span{πn|n∈N}. AK-space which is a Fréchet (Banach) space is called anFK-(BK-)space. The sequence spaces

m:=

x∈ω| x∞:=sup

k

xk<∞

, c:=

x∈ω| xk

converges, that is limx:=lim

k xkexists

, c0:=

x∈c|limx=0 , cs:= x∈ω

k

xkconverges

,

:= x∈ω

k

xk<∞

,

bv:= x∈ω

k

xk−xk+1<∞

,

(2.1)

(together with their natural norm) are important as well as well-known examples of

BK-spaces. Furthermore,ωis anFK-space where its (unique)FK-topology is generated by the family of semi-normsrk,rk(x):= |xk|(x∈ω, k∈N).

For sequence spacesEandF we use the notation E·F:=

yx:= ykxk

|y∈E, x∈F , E^F:=

y∈ω| ∀x∈E:yx∈F

. (2.2)

In this way, the well-knownα-dualE^αandβ-dualE^βofEare defined asE^α:=Eand E^β:=E^cs, respectively.

IfEis aK-space that containsϕ, then E^f:=

uf:=

f e^k

|f∈E

(2.3) is calledf-dual ofE. (Note, throughout we will use the notationup:=(p(e^k))for each functionalp:E→K.) Moreover, we put

ESAK:= x∈E| ∀f∈E:f (x)=

k

xkf e^k

,

EUSAK:= x∈E| ∀f∈E:f (x)=lim

Ᏺ∈Φ

k∈Ᏺ

xkf e^k

,

(2.4)

whereΦ is the set of all finite subsets ofNdirected by “set inclusion” (cf. [18,19]).

AK-spaceEis called aSAK-(USAK-)space ifE=ESAK(E=EUSAK). If ¯ϕ=E, thenEis anAD-space by definition.

LetA=(ank)be an infinite matrix. For a sequence spaceEwe call EA:= x∈ω|Ax:=

k

ankxk

n

exists andAx∈E

(2.5) domain ofA(relative toE). IfEis a (separable)FK-space, thenEAis too. In particular, the domaincA= {x∈ωA|limAx:=limAxexists}is a separableFK-space.

Obviously,ϕ⊂cAif and only ifak:=limnankexists for everyk∈N.Ais called an Sp₁-matrix ifak=1(k∈N), and an Sp^∗₁-matrix if, in addition, each column of Abelongs tobv. IfE and F are sequence spaces withE⊂FA, then the linear map A:E→F,x→Axis called matrix map.

LetEbe a linear space. For a subsetMofE^∗we use the following notation:

M:=

g∈E^∗| ∃ gn

inMsuch thatgn →g σ

E^∗, E ,

M:=

L⊂E^∗|Lis a linear subspace ofE^∗andM⊂L=L

, M^b:=

g∈E^∗| ∃ gα

α∈ᏭinMsuch that gα|α∈Ꮽisσ

E^∗, E

-bounded andgα →g σ

E^∗, E , M^b:=

L⊂E^∗|Lis a linear subspace ofE^∗andM⊂L=L^b

.

(2.6)

Following [5,6], aK-space E is called anLϕ-space and an Aϕ-space, if E⊂ϕand E⊂ϕ^b, respectively. Note,τ(E, ϕ) andτ(E, ϕ^b)is, respectively, the strongestLϕ- topology andAϕ-topology on an arbitrarily given sequence spaceE.

Theorem2.1(see [6, Theorems 3.2 and 3.9]; see also [4,5]). LetF be aK-space.

(a)Fis anLϕ-space if and only if for each MackeyK-spaceEwithσ (E, E)-sequentially complete dual each matrix mapA:E→F is continuous.

(b)F is anAϕ-space if and only if for every barrelledK-spaceE each matrix map A:E→F is continuous.

Theorem2.2(see [6, Theorem 4.8]; see also [4]). LetA=(ank)be a matrix. IfEis anyLϕ-space (Aϕ-space), thenEA(endowed with its natural topology) is anLϕ-space (Aϕ-space).

3. Dual pairs(E, E^S). Throughout, let(S, τS)be aK-space containingϕwhereτS

is generated by a familyᏽ of semi-norms, and, moreover, lets∈Sbe a sum onS (cf. [17]), that is,

s(z)=

k

zk for eachz∈ϕ. (3.1)

Furthermore, letEbe a sequence space containingϕ. Then(E, E^S)is a dual pair where its bilinear form,is defined byx, y:=s(yx)for allx∈E,y∈E^S; thereforeE^S⊂ E^∗(up to isomorphy where the isomorphismE^S→E^∗is given byy→s◦diag_y:E→K and diag_y is the diagonal matrix (map onE) defined by u). Because ofϕ⊂E^S, the weak topologyσ (E, E^S)is aK-topology. In case of

S:=cs, s(z):=

k

zk:=lim

n

n k=1

zk (z∈cs), (3.2)

S:=, s(z):=lim

F∈Φ

k∈F

zk (z∈), (3.3)

we get the dual pairs(E, E^β)and(E, E^α), respectively, which play a fundamental role in summability and the study of topological sequence spaces.

Obviously,E^β⊂EifEis aK-space and(E, σ (E, E))is sequentially complete. For example, the latter holds for all barrelledK-spaces.

In view of this remark it is natural to ask for sufficient conditions in order that the inclusionE^S⊂Eholds (up to isomorphy). Aiming an answer to this question we mention (cf.Theorem 2.1) that a matrix map A:E →S is continuous if one of the following conditions occurs:

(A)S is anLϕ-space,Eis a K-space equipped with the Mackey topologyτ(E, E), and(E, σ (E, E))is sequentially complete.

(B)Sis anAϕ-space andEis a barrelledK-space.

In particular, (A) as well as (B) implies for eachy∈E^Sthe continuity of the matrix map diag_y:E→S. Thus we have the following proposition.

Proposition3.1. IfEas well asSenjoy one of the statements (A) or (B), thenE^S⊂E. Remark 3.2. Let E be a sequence space withϕ⊂E. One may easily check that (ϕ, σ (ϕ, E)) is sequentially complete, and (E, τ(E, ϕ^b))is barrelled. As immediate consequences ofProposition 3.1we obtain thatE^S⊂ϕfor eachLϕ-spaceSandE^S⊂ ϕ^bfor eachAϕ-spaceS.

Definition3.3. For aK-spaceEcontainingϕwe put ESK:=

x∈E| ∀f∈E: ufx

∈Sandf (x)=s ufx

. (3.4)

Eis called anSK-space ifE=ESK.

Remark3.4. (a) IfEis aK-space containingϕ, thenESK⊂ϕinE.

(b)(E, τ(E, E^S))is anSK-space for each sequence spaceEwithE⊃ϕ.

The latter remark is an immediate consequence of the following result which will be useful in the sequel:LetEbe a sequence space containingϕandF be aK-space withE⊂F. If the inclusion mapi:(E, τ(E, E^S))→Fis continuous, thenE⊂FSK.

Remark3.5. (a) In the particular case of (3.2) and (3.3) theSK-property is identical withSAK andUSAK, respectively.

(b) Clearly, ifEis anSK-space, thenEis anAD-space andE^f⊂E^S. Conversely, if (A) or (B) holds, thenE^f ⊂E^S forcesEto be anSK-space. Indeed, in the latter situation s◦diag_u

f∈Eand the equationf=s◦diag_u

f extends fromϕtoE.

4. The solid topology. A sequence spaceEis solid provided thatyx∈Ewhenever y∈mandx∈E. In this situationx∈Eif and only if|x|:=(|xk|)∈E.

Motivated by Große-Erdmann [12] we introduce some notation.

Notation4.1. Under the assumption thatSis solid for aK-spaceEcontainingϕ we put

ESC:=

x∈E| ∀p∈ᏼE:upx∈S andp(x)≤s

up|x|

, (4.1)

whereᏼEdenotes the family of all continuous semi-norms onE. IfE=ESC, thenEis called anSC-space.

In the particular case of (3.3) we get (cf. [12, page 502])

ESC=ACE:= x∈E| ∀p∈ᏼE:

k

p xke^k

<∞

. (4.2)

We assume throughout this section thatS is solid and the sums∈S is a positive functional, that is,

s(z)≥0 for eachz∈Swith zk≥0(k∈N). (4.3) An important example for this situation is given in (3.3). We are going to present a further one.

Example4.2. LetT=(tnk)be a normal Sp₁-matrix such thattnk≥0(n, k∈N). We put

S:= z∈mT| z[T ]:=sup

n

k

tnkzk<∞

. (4.4)

As we may easily check,(S, [T ])is a solidBK-space (containingϕ). Now, we will show that there exists a positive sumsonS.

First of all, we note that(m, _∞,≥)and (mT, T,≥T)with T := ◦T are equivalent as ordered normed spaces, wherez≥0 is defined byzk≥0(k∈N)and z≥T0 by T z≥0. Sincee:=(1,1, . . .)is an interior point of the positive coneK:=

{z∈m|z≥0}in(m, _∞), we get thatT⁻¹eis an interior point of the positive cone KT:= {z∈mT|z≥T0}in(mT, T). By a result of Krein (cf. [22, Theorem XIII.2.3]), each positive functionalg∈(cT, T)can be extended to a positive continuous linear functional on(mT, T). In particular, there exists ans∈(mT, T)such that

s|cT=limT, s(z)≥0 z∈KT

. (4.5)

Thens:=s|S∈(S, [T ])ands(z)≥0 for allz∈S∩KT. Because ofK⊂KT we have (4.3), andsis a sum sinceT is an Sp₁-matrix.

Definition4.3. (a) LetEbe a sequence space andGa subspace ofE^Scontainingϕ.

The locally convex topologyν(E, G)onE generated by the semi-normsp_|y|(y∈G) with

p_|y|(x):=s

|yx|

(x∈E) (4.6)

is called thesolid topologycorresponding to the dual pair(E, G).

(b) Note thatν(E, G)is the topologyτS of uniform convergence on the solid hulls ofy ∈G, that is,ν(E, G)=τS with S:= {sol{y} |y ∈G}and sol{y}:= {v∈ω|

|vk| ≤ |yk|(k∈N)}. In fact, it is not difficult to verify thatp_|y|(x)=sup{|s(vx)| | v∈sol{y}}. Therefore,(E, ν(E, G))= |G|(the solid span ofG).

(c) AK-space(E, τE)is called simple if eachτE-bounded subsetDofEis dominated by anx∈E, that is,D⊂sol{x}.

Proposition4.4. LetEbe a sequence space containingϕ.

(a)The solid topologyν(E, E^S)is compatible with(E, E^S), that is,(E, ν(E, E^S))=E^S. (b)(E, ν(E, E^S))is an SK-space.

(c)(E, ν(E, E^S))SCis an SC-space.

Proof. The statement (a) follows fromRemark 4.3(b) and the fact thatE^Sis solid.

Obviously, byRemark 3.4(b) we get that (a) implies (b).

(c) Letpbe a continuous semi-norm on(E, ν(E, E^S)), thenp(x)=sup_y∈V|s(yx)| for eachx∈E, whereV := {y∈E|p(y)≤1}^◦ (polar in E^S). There exists v∈E^S such thatp(x)≤p_|v|(x)for allx∈E, hencep(e^k)≤ |vk|(k∈N). SinceE^S is solid andv∈E^S, then up=(p(e^k))∈E^S. Furthermore,p(e^k)=sup_y∈V|yk|(k∈N)and

|s(yx)| ≤s(|yx|)≤s(up|x|) (x∈E)for everyy∈V, that is,p(x)≤s(upx)for eachx∈E.

The following proposition is in the particular case of (3.3) a result due to Große- Erdmann (cf. [12, Theorem 3.2] and the erratum in [13]).

Proposition4.5. LetEbe a solid sequence space containingϕandGbe a sequence space withϕ⊂G⊂E^S. Then the following statements are equivalent:

(a)(E, σ (E, G))is simple.

(b)(G, ν(G, E))is barrelled.

If in additionGis solid, then (a), thus (b), is equivalent to (c)(E, ν(E, G))is simple.

Proof. (a)⇒(b). If(E, σ (E, G))is simple, then for eachσ (E, G)-bounded subsetBof Ethere existsy∈Esuch thatB⊂sol{y}. Thusβ(G, E)⊂ν(G, E), which is equivalent to the barrelledness of(G, ν(G, E)).

(b)⇒(a). SupposeBisσ (E, G)-bounded subset ofE. Then the semi-normpdefined by p(y):=supx∈B|s(yx)|(y∈G)is continuous on(G, β(G, E)), which forcesβ(G, E)= ν(G, E)by (b). Hencep(y)≤sup_x∈sol{v}|s(yx)|(y∈G)for a suitablev∈E. Putting y:=e^k(k∈N)we get sup_x∈B|xk| =p(e^k)≤sup_x∈sol{v}|xk| = |vk|(k∈N). Thus,B is dominated byv.

IfGis solid, thenν(E, G)- andσ (E, G)-boundedness of subsets ofEare obviously equivalent, thus “(a)(c)” holds.

5. Quasi-matrix maps. Aiming the extension of some well-known inclusion theo- rems due to Bennett and Kalton, we consider in the sequel the so-called quasi-matrix maps.

Definition5.1. LetS be a K-space with a sum s∈Sand let A=(ank)be an infinite matrix anda⁽ⁿ⁾itsnth row. Then the linear map

A:E →ω, x→Ax:= s

a⁽ⁿ⁾x

n, (5.1)

whereEis a linear subspace of the sequence spaceωA:=_∞

n=1{a⁽ⁿ⁾}^S, is calledquasi- matrix map. Moreover, for every sequence spaceF the sequence space

FA:=

x∈ωA|Ax∈F

(5.2) is called adomain ofArelative toFandcAis called thedomainofA; ifx∈cA, we put

limAx:=lim

n s a⁽ⁿ⁾x

. (5.3)

Note,Ax=Axif the matrixAis row-finite.

First of all we give sufficient conditions for the continuity of quasi-matrix maps.

Theorem5.2. LetEandF beK-spaces. Each of the following conditions implies the continuity of each quasi-matrix mapA:E→F:

(A) S andF areLϕ-spaces,E is a Mackey space, and(E, σ (E, E))is sequentially complete.

(B) SandF areAϕ-spaces andEis barrelled.

Proof. LetA:E→F be a quasi-matrix map defined by (5.1). We put∆A:= {f∈ F |f◦A ∈ E} and show ∆A =F; then A is weakly continuous, hence continu- ous since E carries the Mackey topology τ(E, E). First of all we note ϕ⊂∆_A. In- deed, if w = (w1, . . . , wn₀,0, . . .)∈ ϕ and f (y) :=

nwnyn (y ∈F ), then we get f◦A(x)=n0

n=1wns(a⁽ⁿ⁾x)=n0

n=1s(wna⁽ⁿ⁾x)=s(vx),x∈E, wherev:=w1a⁽¹⁾+

···+wn₀a⁽ⁿ⁾∈E^S. Thusf◦A∈EbyProposition 3.1.

The case(A). Since(E, σ (E, E))is sequentially complete, then∆A∩F=∆A. There- fore, the fact thatF is anLϕ-space impliesF=ϕ∩F⊂∆A∩F=∆A.

The case(B). The barrelledness ofEimplies∆A

b∩F=∆A. SinceFis anAϕ-space, we haveF=ϕ^b∩F⊂∆A

b∩F=∆A.

Now, we are going to define aK-topology on the domainE_A, whereE is aK-space topologized by a familyᏼof semi-norms andAis a quasi-matrix map defined by (5.1).

The setQn:= {rk|k∈N}∪{q◦diaga⁽ⁿ⁾|q∈ᏽ}of semi-norms generates aK-topology on{a⁽ⁿ⁾}^S (n∈N). IfS is anLϕ- orAϕ-space, then theK-space{a⁽ⁿ⁾}^S enjoys the same property. Clearly, the familyQ:=

n∈NQngenerates aK-topologyτω_AonωA. By the following lemma, theLϕ- andAϕ-property of theK-spaceS implies the same property on(ω_A, τω_A).

Lemma5.3. LetS be anLϕ-space (Aϕ-space) and let {Fα|α∈Ꮽ}be a family of Lϕ-spaces (Aϕ-spaces). Then᏾:=

α∈Ꮽ᏾α, where᏾α is a family of semi-norms on Fα(α∈Ꮽ)generating the topology ofFα, generates anLϕ-topology (Aϕ-topology) on F:=

α∈ᏭFα.

Proof. First assume thatSandFα(α∈Ꮽ)areLϕ-spaces and(E, τE)is a Mackey K-space such that(E, σ (E, E))is sequentially complete. Moreover, letA:E→F be a matrix map. Because ofTheorem 2.1(a) the mapA:E→Fα is continuous for each α∈Ꮽ. This implies the continuity ofA:E→F. ByTheorem 2.1(a),F is anLϕ-space.

For the case ofAϕ-spaces we useTheorem 2.1(b).

Proposition5.4. IfEandSareLϕ-spaces (Aϕ-spaces), then(EA, τE_A)topologized by the familyQ∪{p◦A|p∈ᏼ}of semi-norms is also anLϕ-space (Aϕ-space).

Proof. Obviously,EA is aK-space. First of all we verify that (EA, τE_A)is anLϕ- space: ifE and S areLϕ-spaces, thenωA is also an Lϕ-space. By Theorem 5.2, the maps

iω: EA, τ

EA, ϕ

→ωA, x→x, A: EA, τ

EA, ϕ

→E, x→Ax, (5.4)

are continuous. This implies the continuity of the identity mapi:(EA, τ(EA, ϕ))→ (EA, τE_A). Since(E, τ(EA, ϕ))is anLϕ-space we get that(EA, τE_A)is anLϕ-space too.

For the case ofAϕ-spaces we use the fact that(EA, τ(EA, ϕ^b))is anAϕ-space.

Remark 5.5. If E is a separable FK-space and S is a separable BK-space, then (EA, τE_A)is a separableFK-space. The proof of this fact is similar to the one of [1, Theorem 1].

ByProposition 5.4, the domaincAof a quasi-matrix mapAis anLϕ- orAϕ-space if S has the same property. Ifak:=limnankexists, thenϕ⊂cA, and we puta:=(ak) and define

Λ^S_A^⊥:=

x∈cA|ax∈Sand limAx=s(ax)

. (5.5)

Note, because of limA∈c_A, we have(cA)SK⊂Λ^S⊥_A .

6. Inclusion theorems of Bennett-Kalton type. In this section, we extend well- known inclusion theorems of Bennett and Kalton [3, Theorems 4 and 5] (cf. also au- thors’ results [5, Theorem 4.4] and [6, Theorem 5.1]) to the situation considered above.

We extend moreover a theorem due to Große-Erdmann (cf. [12]) which is of the same type as the mentioned theorems of Bennett and Kalton. There are no new ideas in the proofs, however for the sake of completeness we give them in a brief form.

Theorem6.1. LetS be anLϕ-space. For any sequence spaceE containingϕthe following statements are equivalent:

(a)(E^S, σ (E^S, E))is sequentially complete.

(b)Each quasi-matrix mapA:(E, τ(E, E^S))→Fis continuous whenF is anLϕ-space.

(c)The inclusion mapi:(E, τ(E, E^S))→F is continuous wheneverF is anLϕ-space containingE.

(d)The implicationE⊂F⇒E⊂FSKholds wheneverF is anLϕ-space.

(e)The implicationE⊂cA⇒E⊂Λ^S⊥_A holds for every quasi-matrix mapA.

Proof. (a)⇒(b) is an immediate consequence ofTheorem 5.2(A), (b)⇒(c) is obvi- ously valid whereas (c)⇒(d) follows fromRemark 3.4(b). Furthermore, “(d)⇒(e)” is true sincecAis anLϕ-space and(cA)SK⊂Λ^S_A^⊥.

(e)⇒(a). If (a⁽ⁿ⁾) is a Cauchy sequence in(E^S, σ (E^S, E)), then E ⊂cA, where the quasi-matrix mapAis defined by the matrixAwitha⁽ⁿ⁾asnth row. On account of (e) we getE⊂Λ^S_A^⊥, thusa∈E^S anda⁽ⁿ⁾→a(σ (E^S, E)).

Theorem6.2. LetS be an Aϕ-space. For any sequence spaceEcontainingϕthe following statements are equivalent:

(a)(E, τ(E, E^S))is barrelled.

(b) Each quasi-matrix map A : (E, τ(E, E^s)) → F is continuous when F is an Aϕ-space.

(c)The inclusion mapi:(E, τ(E, E^S))→F is continuous wheneverF is anAϕ-space containingE.

(d)The implicationE⊂F⇒E⊂FSKholds wheneverF is anAϕ-space.

(e) The implication E⊂mA ⇒E ⊂Λ^S⊥_A holds for every quasi-matrix mapA with ϕ⊂cA.

(f)Everyσ (E^S, E)-bounded subset ofE^S is relatively sequentiallyσ (E^S, E)-compact.

Proof. (a)⇒(b) is an immediate consequence ofTheorem 5.2(B), (b)⇒(c) is obvi- ously valid. (c)⇒(d) follows fromRemark 3.4(b). (d)⇒(e) is true, sincemA is anAϕ- space (cf.Proposition 5.4) and(mA)SK=(cA)SK⊂Λ^S⊥_A .

(e)⇒(f). LetB be aσ (E^S, E)-bounded subset ofE^S and let(b^{(r )})be a sequence in B. Obviously, (b^{(r )}) is bounded in (ω, τω), thus we may choose a coordinatewise convergent subsequence (a⁽ⁿ⁾) of (b^{(r )}). Because of the σ (E^S, E)-boundedness of {a⁽ⁿ⁾|n∈N}we getE⊂mA, whereAis the quasi-matrix map defined by the matrix A=(a⁽ⁿ⁾_k )n,k. Applying (e) we obtainE⊂Λ^S⊥_A , thusa∈E^S anda⁽ⁿ⁾→a(σ (E^S, E)).

HenceBis relatively sequentially compact in(E^S, σ (E^S, E)).

(f)⇒(a). Using the fact that in theK-space(E^S, σ (E^S, E))each relatively sequentially compact subset is relatively compact too (cf. [14, Theorem 3.11, page 61]), (f) tells us that(E, τ(E, E^S))is barrelled.

Motivated by Große-Erdmann [12, Theorem 4.1], we prove next an inclusion theorem which gives a connection between the barrelledness of(E, ν(E, E^S))and the implica- tionE⊂F⇒E⊂FSCwhereF is anAϕ-space.

Theorem6.3. LetS be a solid Aϕ-space with a positive sum s∈S. LetE be any sequence space containingϕ. The following statements are equivalent:

(a)(E, ν(E, E^S))is barrelled.

(b)(E^S, σ (E^S, E))is simple.

(c)Each quasi-matrix mapA:(E, ν(E, E^S))→F is continuous whenF is anAϕ-space.

(d)The inclusion mapi:(E, ν(E, E^S))→F is continuous wheneverF is anAϕ-space containingE.

(e)The implicationE⊂F⇒E⊂FSCholds wheneverFis anAϕ-space.

Proof. (a)(b) is a part ofProposition 4.5, (a)⇒(c) is an immediate consequence of Theorem 5.2(B), (c)⇒(d) is obviously true, and (d)⇒(e) follows fromProposition 4.4(c).

(e)⇒(d). For everyp∈ᏼF we getup∈E^S andp(x)≤p_|up|(x) (x∈E). Thusp|Eis continuous on(E, ν(E, E^S)). Consequently,i:(E, ν(E, E^S))→F is continuous.

(d)⇒(a). First of all we remark that(E, ν(E, E^S))is separable as anSK-space. Thus, we have to show that it isω-barrelled, that is, that every countableσ (E^S, E)-bounded sub- set{a⁽ⁿ⁾|n∈N}ofE^S isν(E, E^S)-equicontinuous (cf. [14, page 27, Theorem 10.2]).

For a proof of this we consider the quasi-matrix mapA defined by the matrixA= (a⁽ⁿ⁾_k )n,k. Then E⊂mA sinceA is σ (E^S, E)-bounded, and(mA, τA) is anAϕ-space because ofProposition 5.4. Hencei:(E, ν(E, E^S))→(mA, τA)is continuous. Further- more, the quasi-matrix mapA:(m_A, τm_A)→(m, _∞)is continuous since _∞◦Ais a continuous semi-norm onmA. Altogether,A:(E, ν(E, E^S))→(m, _∞)is also contin- uous. Thus there existsv∈E^S such thatAx∞=sup_n|s(a⁽ⁿ⁾x)| ≤p_|v|(x)for each x∈E. Consequently, the equicontinuity of{a⁽ⁿ⁾|n∈N}is proved.

7. Applications. In this section, we deal with applications of our general inclusion theorems to the case of certain dual pairs(E, E^S). In various situations we discuss connections between weak sequential completeness and barrelledness on one hand and certain modifications of weak sectional convergence on the other hand.

Case7.1(S=cT andS=cs). LetT=(tnk)be a fixed row-finite Sp1-matrix and S:=cT as well as limTs(z):=z (z∈cT). (7.1) Clearly,Sendowed with its (separable)FK-topology is anLϕ- andAϕ-space. We have (cf. [8,15])

E^S=E^{β(T )}:= y∈ω| ∀x∈E: lim

n

k

tnkykxkexists

(7.2) for each sequence spaceEand

ESK=ESTK:= x∈E| ∀f∈E:f (x)=lim

n

k

tnkxkf e^k

(7.3) (weakT-sectional convergence) ifEis aK-space containingϕ.

Aiming a characterization of quasi-matrix maps in this context we need the follow- ing concept (cf. [7]): letᐂbe a double sequence space, that is a linear subspace of the linear spaceΩof all double sequencesy=(ynr), and letᏭ=(A^{(r )})be a sequence of infinite matricesA^{(r )}=(a^{(r )}_nk)n,k. We put

Ω_Ꮽ:=

r

ω_A(r ), ᐂ_Ꮽ:= x∈Ω_Ꮽ|Ꮽx:=

k

a^{(r )}_nkxk

nr

∈ᐂ

. (7.4)

If there exists a limit functional onᐂ, sayᐂ-lim, then the summability method induced byᐂᏭ and the limit functional

ᐂ- lim_Ꮽ:ᐂ_Ꮽ →K, x→ᐂ- limᏭx, (7.5) is called aᐂ-SM-method. For example, this definition contains as a special case the Ꮿc-SM-methods considered by Przybylski [16] where

Ꮿc:=

y= ynr

n,r∈N

lim

r lim

n ynr=:Ꮿc- limyexists

. (7.6)

Furthermore, if Ꮿm:=

ynr

∈Ω| ∀r∈N: ynr

n∈cand limn ynr

r∈m

(7.7) andᏭ=(A^{(r )})is any sequence of matrices, thenᏯm_Ꮽ:= {x∈Ω_Ꮽ|Ꮽx∈Ꮿm}is an FK-space andᏯc_Ꮽ is a closed separable subspace ofᏯm_Ꮽ.

The introduced notation enables us to describe quasi-matrix maps in the situation of (7.1): ifA=(ank)is any matrix, then the domaincAof the corresponding quasi- matrix mapAis precisely the domainᏯc_Ꮽof theᏯc-SM-methodᏭ=(A^{(r )})withA^{(r )}:= (tnkar k)n,k(r∈N). ThusmA=ᏯmᏭis anAϕ-space andcA=ᏯcᏭ is anLϕ- andAϕ- space (where we consider them asFK-spaces).

As consequences of the main Theorems6.1and6.2we get the following inclusion theorems in the situation of the dual pair(E, E^{β(T )}).

Theorem7.2. For any sequence spaceEwithϕ⊂E the following statements are equivalent:

(a)(E^{β(T )}, σ (E^{β(T )}, E))is sequentially complete.

(b)The implicationE⊂F⇒E⊂FSTKholds wheneverFis anLϕ-space.

(c)The implicationE⊂ᏯcᏭ⇒E⊂(ᏯcᏭ)STKholds for each sequenceᏭof matrices.

Theorem7.3. For any sequence spaceEwithϕ⊂E the following statements are equivalent:

(a)(E, τ(E, E^{β(T )}))is barrelled.

(b)The implicationE⊂F⇒E⊂FSTKholds wheneverFis anAϕ-space.

(c)The implicationE⊂Ꮿm_Ꮽ⇒E⊂(Ꮿc_Ꮽ)STKholds for each sequenceᏭof matrices satisfyingϕ⊂Ꮿc_Ꮽ.

Note, in the particular case ofT :=Σ(summation matrix) we get the well-known inclusion theorems of Bennett and Kalton [3, Theorems 4 and 5] in a generalized version due to the authors (cf. [5, Theorem 4.4] and [6, Theorem 5.1]).

Case7.4(S=bvT). LetT=(tnk)be a fixed row-finite Sp^∗₁-matrix and S:=bvT as well as s(z):=lim

Ᏺ∈Φ

n∈Ᏺ

k

tnk−tn−1,k

zk=limTz

z∈bvT . (7.8)

ThenS equipped with its (separable)FK-topology is anLϕ- andAϕ-space. We have (cf. [10,11])

E^S=E^{α(T )}:= y∈ω

n

k

tnk−tn−1,k

ykxk

<∞

(7.9)

for any sequence spaceE. Moreover, ifEis aK-space containingϕ, then we obtain

ESK=EU ST K:= x∈E| ∀f∈E:

n

k

tnk−t_n−1,k xkf

e^k <∞, f (x)=lim

n

k

tnkxkf

e^k (7.10)

(cf. [11, Theorem 3.1]).

For a description of quasi-matrix maps in the situation of (7.8) we introduce the double sequence spaces

Ꮾvm:= ynr

| ∀r∈N: ynr

n∈bvand

limn ynr

r∈m

, Ꮾvc:=

ynr

| ∀r∈N: ynr

n∈bvand

limn ynr

r∈c

,

(7.11)

and the limit functional Ꮾvc- lim :Ꮾvc →K,(ynr)limrlimnynr. It is a standard exercise to prove thatᏮvm_Ꮽis anFK-space for each sequenceᏭ=(A^{(r )})of matrices and thatᏮvc_Ꮽis a closed separable subspace ofᏮvm_Ꮽ.

For a matrixA=(ank)the corresponding quasi-matrix mapAin the context of (7.8) is theᏮvc-SM-methodᏭ=(A^{(r )})withA^{(r )}:=(tnkar k)nk(r∈N). ThusmA=Ꮾvm_Ꮽ

andcA=Ꮾvc_Ꮽ.

Applying the main Theorems6.1and6.2to the situation of (7.8), we get the follow- ing inclusion theorems.

Theorem7.5. For any sequence spaceEwithϕ⊂E the following statements are equivalent:

(a)(E^{α(T )}, σ (E^{α(T )}, E))is sequentially complete.

(b)The implicationE⊂F⇒E⊂FUSTKholds wheneverF is anLϕ-space.

(c)The implicationE⊂Ꮾvc_Ꮽ⇒E⊂(Ꮾvc_Ꮽ)USTKholds for each sequenceᏭof matrices.

Theorem7.6. For any sequence spaceEwithϕ⊂E the following statements are equivalent:

(a)(E, τ(E, E^{α(T )}))is barrelled.

(b)The implicationE⊂F⇒E⊂FUSTKholds wheneverF is anAϕ-space.

(c)The implicationE⊂Ꮾvm_Ꮽ⇒E⊂(Ꮾvc_Ꮽ)USTKholds for each sequenceᏭof matri- ces satisfyingϕ⊂Ꮾvc_Ꮽ.

Case 7.7(S=). We consider the case of (3.3). Clearly, this is a particular case of (7.8) withT =Σ. ThenE^S=E^αand ESK=EUSAK(cf. [18,19]). A quasi-matrix map induced by a matrixA=(ank)is in this situation a matrix mapyn=

kankxk(n∈N) with the application domainω_|A|:= {x∈ω| ∀n∈N:

k|ankxk|<∞}. We now put m_|A|:=mA∩ω_|A|,c_|A|:=cA∩ω_|A|andΛ^⊥_|A|:= {x∈c_|A||

k|akxk|<∞and limAx=

kakxk}(here we assumeϕ⊂cA). From Theorems6.1and6.2we get the following inclusion theorems.

Theorem7.8. For any sequence spaceEwithϕ⊂E the following statements are equivalent:

(a)(E^α, σ (E^α, E))is sequentially complete.

(b)Every matrix map A:

E, τ E, E^α

→F withE⊂ω_|A| (7.12)

is continuous wheneverFis anLϕ-space.

(c)The implicationE⊂F⇒E⊂FUSAKholds wheneverF is anLϕ-space.

(d)The implicationE⊂c_|A|⇒E⊂Λ^⊥_|A| holds for each matrixA.

Note that Bennett [2] proved, that(E^α, σ (E^α, E))is sequentially complete ifEis a monotone sequence space, and that Swartz and Stuart [21, Theorem 5] (see also [20, Theorem 7]) gave a more general class of sequence spaces having that property.

Theorem7.9. For any sequence spaceEwithϕ⊂E the following statements are equivalent:

(a)(E, τ(E, E^α))is barrelled.

(b)Any matrix map according to (7.12) is continuous whenF is anAϕ-space.

(c)The implicationE⊂F⇒E⊂FUSAKholds wheneverF is anAϕ-space.

(d)The implicationE⊂m_|A|⇒E⊂Λ^⊥_|A|holds for each matrixAsatisfyingϕ⊂cA. Case7.10(S=[cs]). Let

S:=[cs]:= z∈cs lim

j

2^jzk=0

, s(z):=

k

zk

z∈[cs]

, (7.13)

where

2^jak:=2^j+1−1

k=2^j ak(cf. [9]). It is known that[cs]is anAK-BK-space with the norm defined byz:=sup_m|m

k=1zk| +sup_j

2^j|zk|(z∈[cs])(cf. [9, Theorem 4.5]). Thus,[cs]is anLϕ- andAϕ-space. For a sequence spaceEwe setE^[β]:=E^[cs]= {y∈E^β|limj

2^j|ykxk| =0}. IfEis aK-space containingϕ, we get ESK= x∈E| ∀f∈E:

ufx

∈[cs]andf (x)=

k

xkf e^k

= E^f[β]

∩ESAK.

(7.14)

The quasi-matrix mapA:ωA→ωcorresponding to a matrixA=(ank)is precisely the matrix mapA:ω[A]→ωwith

ω[A]:= x∈ωA| ∀n∈N: lim

j

2^j

ankxk=0

=ωA. (7.15)

We putm[A]:=mA∩ω[A],c[A]:=cA∩ω[A]and, assumingϕ⊂cA, Λ^⊥_[A]:= x∈c[A]|ax∈[cs]and lim

A x=

k

akxk

. (7.16)

From Theorems6.1and6.2we obtain the following inclusion theorems in the con- text of (7.13).

Theorem7.11. For any sequence spaceEwithϕ⊂Ethe following statements are equivalent:

(a)(E^[β], σ (E^[β], E))is sequentially complete.

(b)Each matrix map A:

E, τ E, E^[β]

→F withE⊂ω[A] (7.17)

is continuous wheneverFis anLϕ-space.

(c)The implicationE⊂F⇒E⊂(F^f)^[β]∩FSAKholds wheneverF is anLϕ-space.

(d)The implicationE⊂c[A]⇒E⊂Λ^⊥_[A]holds for each matrixA.

Theorem7.12. For any sequence spaceEwithϕ⊂Ethe following statements are equivalent:

(a)(E, τ(E, E^[β]))is barrelled.

(b)Each matrix map according to (7.17) is continuous whenF is anAϕ-space.

(c)The implicationE⊂F⇒E⊂(F^f)^[β]∩FSAKholds wheneverF is anAϕ-space.

(d)The implicationE⊂m[A]⇒E⊂Λ^⊥_[A]holds for each matrixA.

Acknowledgement. The work presented here was supported by Deutscher Akad- emischer Austauschdienst (DAAD) and Estonian Science Foundation Grant 3991.

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Johann Boos: Fachbereich Mathematik, FernUniversität Hagen, D-58084 Hagen, Germany

E-mail address:johann.boos@fernuni-hagen.de

Toivo Leiger: Puhta Matemaatika Instituut, Tartu Ülikool, EE50090Tartu, Estonia E-mail address:leiger@math.ut.ee