LOCAL COMPLETENESS OF

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LOCAL COMPLETENESS OF

(E), 1 ≤ p < ∞

C. BOSCH, T. GILSDORF, C. GÓMEZ, and R. VERA Received 10 September 2001 and in revised form 26 February 2002

We study the heredity of local completeness and the strict Mackey convergence property from the locally convex spaceEto the space of absolutelyp-summable sequences onE, p(E)for 1≤p <∞.

2000 Mathematics Subject Classification: 46A03, 46A17.

1. Introduction. In 1956, Grothendieck [5], introduced the Banach-valued sequence space p(E), the space of absolutelyp-summable sequences on a Banach spaceE, where he discussed tensor products ofp and E, with 1≤ p≤ ∞. Later, in 1969 Pietsch [8] used Banach-valued sequence spacesp(E), to studyp-summing opera- tors between Banach spaces, also see Diestel et al. [2]. In this paper, we discuss how local completeness and the strict Mackey convergence condition ofEimply local com- pleteness and the strict Mackey convergence condition inp(E)in the case 1≤p <∞. The casep= ∞was studied in [1].

2. Definitions and notation. Throughout this paper,(E,t)denotes a Hausdorff lo- cally convex space overK (RorC) and {ρj}j∈J denotes the family of continuous seminorms associated with the topologytonE.

LetD⊂Ebe a bounded, closed, and absolutely convex set. Denote byED= ∪^∞_k=1kD, and for eachx∈ED,ρD(x)=inf{r >0 :x∈r D}, the Minkowski seminorm associated with D. Now ED⊂E and the boundedness ofD implies thati:(ED,ρD)→(E,t)is continuous, andρDis a norm so that, for everyj∈Jthere existsrj∈R⁺such that ρj|ED≤rjρD.

Remark2.1. For eachD⊂Ebounded, closed, and absolutely convex, the family of seminorms{ρj}j∈Jcan be replaced by an equivalent family{ρ_j}j∈Jsuch thatρ_j≤ρD. To construct the family{ρ_j}j∈J we know that there existsrj>0 such thatρj(x)≤ rjρD(x)for everyx∈EDso it suffices to takeρ_j=(1/rj)ρjifrj>1, and we will have ρ_j≤ρD, for everyj∈J. For simplicity we will always work with an equivalent family of seminorms, also denoted by{ρj}j∈Jsuch thatρj(x)≤ρD(x)holds for everyj∈J andx∈ED.

A bounded, closed, and absolutely convex setD⊂E, called a disk, is a Banach disk if(ED,ρD)is a Banach space. If every bounded setA⊂Eis contained in a Banach disk we say thatE is locally complete. Let(E,t)satisfies the strict Mackey convergence condition if for every bounded setA⊂E, there exists a diskDthat containsAsuch that the topologies of(E,t)and(ED,ρD)agree onA.

Every metrizable space satisfies the strict Mackey convergence condition, [7]. In addition, the strict Mackey convergence condition is preserved under the formation of closed subspaces, countable products, and countable direct sums, [6]. The strict Mackey convergence condition for webbed spaces is studied in [3,4].

Remark2.2. Using the family of seminorms{ρj}j∈Jit is easy to see that the strict Mackey convergence condition is equivalent to: for eachDthere existsj0∈J such thatρj0|D=ρD.

Let p be a real number such that 1≤p <∞. The spacep(E) of absolutely p- summable sequences onEis

p(E)=

xn

⊂E: ∞ n=1

ρ^p_j xn

<∞,∀j∈J

. (2.1)

The family of seminormsρρ_j((xn))=(_∞

n=1ρ_j^p(xn))^1/p,j∈J, induce a topology of locally convex space inp(E); we will denote byτthis topology.

The spacep(ED)is defined byp(ED)= {(xn)⊂ED:_∞

n=1ρ^p_D(xn) <∞}and en- dowed with the topology generated by the norm

ρρD

xn

= _∞

n=1

ρ^p_D xn1/p

. (2.2)

We denoteAD= {(xn)∈p(E):(xn)n∈N⊂D}.

Note thatρρ_j|p(ED)≤ρρD for everyj∈Jsinceρj|ED≤ρD.

3. Bounded sets. In this section, we characterize the bounded sets ofp(E)in terms of the bounded sets ofE.

Lemma3.1. LetDbe a disk in(E,t); then

(i) p(ED)⊆ {(xn)∈p(E):{xn} ⊂kDfor somek∈N};

(ii) if there existsj0∈J, depending onD, such thatρj0|D=ρD(i.e., the strict Mackey convergence condition holds), then {(xn)∈p(E):{xn} ⊂kDfor somek∈ N} ⊂p(ED).

Proof. (i) Let(xn)∈p(ED). Then_∞

n=1[ρD(xn)]^p<∞so that givenε=1 there existsn0∈N, such that for eachn > n0, we haveρD(xn)≤(_∞

n0ρ_D^p(xn))^1/p≤1 which means thatxn∈Dfor everyn > n0.

Now for i= 1,2,...,n0 there exists ki ≥ 0 such that xi ∈ kiD. We take k = max{1,k1,...,kn0}. Then{xn} ⊂kD and we have p(ED)⊂ {(xn)∈p(E):{xn} ⊂ kDfor somek∈N}.

(ii) Let(xn)∈ {(yn)∈p(E):{yn} ⊂kDfor somek∈N}. Thusxn∈EDfor every n∈Nsince{xn} ⊂kD.

Now observe that_∞

n=1ρD^p(xn)=_∞

n=1ρ^p_j0(xn) <∞since(xn)∈p(E). Hence in this case we have the equalityp(ED)= {(xn)∈p(E):{xn} ⊂kDfor somek∈N}.

Remark3.2. Note thatkAD=AkDfor everyk∈N.

Corollary 3.3. If E satisfies the strict Mackey convergence condition, then p(E)AD=p(ED).

Proof. It follows from the equality in the proof ofLemma 3.1(ii) thatp(E)AD ⊂ p(ED). Now let(xn)∈p(ED). Then byLemma 3.1(i),(xn)⊂kDfor somek∈Nso {xn} ⊂AkD=kADand(xn)∈p(E)AD.

Remark3.4. If(E,t)satisfies the strict Mackey convergence condition, then p(E)AD=p

ED

= xn

∈p(E): xn

⊂AkDfor somek∈N

. (3.1)

Lemma3.5. (i)ρAD((xn))=sup{ρD(xn):n∈N};

(ii) ρAD((xn))≤ρρD((xn))for every(xn)∈p(ED).

Proof. (i) Lets=sup{ρD(xn):n∈N}. Then{xn} ⊂sDso {xn} ⊂AsD =sAD

and thenρAD((xn))≤s. Now taker=ρAD((xn)). Then{xn} ⊂r AD=Ar Dand then {xn} ⊂r Dwhich means thatr≥s.

(ii) ρρD((xn))=(_∞

n=1ρ_D^p(xn))^1/p ≥ρD(xn) for every n∈N. Using (i) we have ρρD((xn))≥ρAD((xn)).

Note thatADis not bounded inp(E); we need to construct a “smaller” set, in the sense of boundedness.

Define for eachj∈Jandm∈Nthe setAD(j,m)= {(xn)n∈AD:ρρ_j((xn))≤m}

and for eachB⊂p(E), letB^∗= {x∈E:x∈ {xn}and(xn)∈B}.

The next proposition gives a way to look at the bounded sets inp(E).

Proposition 3.6. If β= {Dλ}λ∈∧ is a fundamental system of bounded disks in E, then {C = ∩j∈J{AD_λ(j,mj)}:λ∈Λ, (mj)∈N^J}is a fundamental system of τ- bounded sets inp(E).

Proof. LetB⊂p(E)be a bounded set. ThenB^∗is bounded inEsoB^∗⊂Dλfor some λ. For each x∈B^∗, if x∈(xn)then givenj ∈J there is some sj such that ρj(x)≤ρρj((xn))≤sjso thatρρj(B)≤s_j. Now letmj∈Nbe such thatsj≤mj. We haveB⊂C=∩j∈JAD_λ(j,mj).

Remark3.7. (i) IfDis bounded inE, then for eachj∈J, byRemark 2.1ρ_j|ED≤ρD. (ii) IfCis bounded inp(E), then for eachj∈J, byRemark 2.1ρρj|p(E)_C ≤ρ_C. 4. Main results

Proposition4.1. If for someDthere existsj0∈J, such thatρj0|D=ρDinE, then ρρj0|C =ρC whereC = ∩j∈JAD(j,mj)inp(E). Equivalently, if E satisfies the strict Mackey convergence condition, thenp(E)also satisfies the strict Mackey convergence condition.

Proof. Let(xn)∈C. Thens=ρρj0(xn)=(_∞

n=1ρ_j^p₀(xn))^1/p=(_∞

n=1ρ^p_D(xn))^1/p≥ ρD(xn)≥ ρρ_j(xn) for every j ∈J and n∈N. So we have (xn)∈ ∩j∈JAD(j,s)= s[∩j∈JAD(j,1)]⊂sC. ThusρC((xn))≤s=ρρj0(xn)and sinceCis bounded inp(E) we haveρρj ≤ρC for eachj∈J; now ρρj|C ≤ρC for everyj∈J, so forj0we have ρρ_j0|C=ρC.

Notice that ifBis a bounded set inp(E), thenρρ_j(B)≤mjfor allj∈Jwithmj∈N and thenB⊂ ∩j∈JAB^∗(j,mj).

This gives the property we need to characterize the bounded sets inp(E). Theorem4.2. IfEis locally complete and satisfies the strict Mackey convergence condition, then(p(E)C,ρC)whereC= ∩j∈JAD(j,mj)inp(E), is a Banach space so p(E)is locally complete.

Proof. LetDbe a bounded closed disk such that(ED,ρD)is a Banach space and let C = ∩j∈JAD(j,mj). By Remark 2.1 there is a j0∈ J such that ρj0|D =ρD. We will show that (p(E)C,ρC)is a Banach space. ByCorollary 3.3 we havep(E)AD = p(ED)and sinceC ⊂AD,p(E)C ⊂p(E)A_D. Let(x_n^k)k∈N⊂p(E)C be aρC-Cauchy sequence. Thus for everyε >0 there existsN∈Nsuch that for everyn,m≥Nwe have ρC((x_n^k)−(x_m^k)) < ε. UsingRemark 3.7(ii) we have thatρρj|(E)C≤ρC. Hence(x^k_n) is also aρρ_j-Cauchy sequence and then aρρ_j0-Cauchy sequence. ThusρD(x^kn−x^km)= ρj0(x_n^k−x_m^k)≤ρρ_j0((x^k_n)−(x^k_m)), then the sequence(x^k_n)k∈Nfor everyn∈Nis also aρD-Cauchy sequence in(ED,ρD)which is a Banach space, so there existsz^kinED

such that(x^k_n)converges toz^kwith respect to the normρD. UsingRemark 3.7(i) we haveρj|ED≤ρD. Hence, we have the following claims.

Claim1. We have that(x^k_n)converges toz^kwith respect to the seminormρjfor everyj∈J.

Claim2. Consider the sequence formed by the(z^k)k∈N∈p(ED). We compute ∞

k=1

ρD z^k_p

= lim

m→∞

m k=1

ρD z^k_p

= lim

m→∞

m k=1

ρj0

z^kp

= lim

m→∞

m k=1

ρj0

lim

n→∞x_n^k_p

= lim

m→∞lim

n→∞

m k=1

ρj0

x_n^kp

≤ lim

m→∞lim

n→∞

∞ k=1

ρj0

x_n^k_p

=lim

n→∞

∞ k=1

ρj0

x_n^kp

≤lim

n→∞ρρ_j0 xn

≤ε+ρρ_j0 xN

<∞, for someN∈N.

(4.1)

In this last inequality we usedxn=(x_n^k)k∈Nand since it is aρρ_j0-Cauchy sequence, given ε >0, ρρj0(x_n^k)−ρρj0(x^k_m)≤ρρj0((x^k_n)−(x^k_m)) < ε for every n,m > N, so ρρ_j0((xn))≤ε+ρρ_j0((xN)). Notice that(xn)is aρρ_j-Cauchy sequence for everyj∈J.

Therefore forj0and consequently forρρD, then for everyε >0 there is anN∈Nsuch thatρD(x^kn−z^k)=ρD(x^kn−lim_m→∞xm^k)=lim_m→∞ρD(xn^k−xm^k) < ε.

Claim3. The sequence(x^k_n)converges to(z^k)k∈Ninp(ED). Since

ρρD

x^k_n− z^k

k

=



^∞

k=1

ρ^p_D

x^k_n−z^k



1/p

≤



^N

k=1

ρ^p_D

x^k_n−z^k +ε^p

2





1/p

≤





ε^p

2N+···+ε^p 2N

Nfactors

+ε^p 2







1/p

=ε, forn > N.

(4.2)

In the first inequality we usedClaim 2. This completes the proof of the convergence.

Claim 4. We have (z^k)k∈N ∈ p(E)C. (x_n^k)k∈N is a ρC-Cauchy sequence so it is bounded and there is ans∈Nsuch that(x_n^k)⊂sC. UsingClaim 3,(x_n^k)converges to(z^k)inp(E)C with respect toρρD and sinceρρj|p(ED)≤ρρD for everyj∈Jthe sequence(x_n^k)is τ-convergent to(z^k), it is convergent for eachρρj. Now for each ε >0 there existsNjsuch thatρρ_j((z^k))≤ρρ_j((z^k)−(x_n^k))+ρρ_j((x^k_n)) < ε+smjfor everyj∈Jandn≥Nj, this means that(z^k)∈sC⊂p(E)C.

Claim5. The sequence(x_n^k)converges to(z^k)k∈Ninp(E)C. Letε >0, since(x_n^k)is aρC-Cauchy sequence, there isN∈Nsuch that(x_n^k)−(x_m^k)∈εCfor everyn,m≥N.

C is τ-closed so(x_n^k)−(τ−lim(x^k_m))∈εC; then(x_n^k)−(z^k)∈εC for everyn≥N which meansρC((x_n^k)−(z^k))≤εfor everyn≥N.

Notice that this is true for every 1≤p <∞. The casep= ∞also follows from this and we get the characterization given in [1], although under a stronger hypothesis.

Here we needEto satisfy the strict Mackey convergence condition.

Lemma4.3. If D⊂E ist-complete and the net{xλ}Λis a τ-Cauchy net bounded with respect toρC, that is if there existss∈Nsuch that{xλ}Λ⊂sC then there exists z∈2sCsuch thatxλconverges tozwith respect to theτtopology inp(E).

Proof. Let{xλ}Λbe aτ-Cauchy net,xλ=(x_λ¹,x_λ²,...), then for everyε >0 there exists λj∈Λsuch that for every j ∈J, ρj(x^k_λ−x_λ^k)≤ρρj(xλ−xλ) < ε for every λ,λ≥λjandk∈N. So{x_λ^k}Λ⊂Dist-Cauchy for eachk∈N, and sinceDis complete there is az^ksuch thatx_λ^kconverges toz^kwith respect to the topologytfor eachk∈N. Letz= {z¹,z²,...}. Thenz⊂D, and for eachj∈Jandk∈Nwe haveρj(x_λ^k−z^k)= ρj(x_λ^k−(ρj−lim_λx_λ^k))=lim_λρj(x_λ^k−x^k_λ), so raising to thepth power and adding with respect tokwe have

∞ k=1

ρj

x^k_λ−z^k_p

= lim

n→∞

n k=1

ρj

x_λ^k−z^k_p

= lim

n→∞

n k=1

limλ ρj

x^k_λ−x_λ^k_p

= lim

n→∞lim

λ

n k=1

ρj

x_λ^k−z^k_p

≤lim

λ

∞ k=1

ρj

x_λ^k−z^kp

=lim

λ ρρ_j

xλ−xλ

< ε^p,

(4.3) for everyλ≥λj.

So we haveρρj(xλ−z)^p=_∞

k=1ρj(x_λ^k−z^k)^p< ε^p, for everyλ≥λj. This means that xλconverges tozwith respect to the topologyτ. We still need to prove thatz∈p(E)

ρρj(z)^p= ∞ k=1

ρj z^kp

= ∞ k=1

ρj

z^k+x_λ^k−x^k_λp

≤ ∞ k=1

2^p ρj

z^k−x^k_λp+ρj x^k_λp

=2^p ∞ k=1

ρj

z^k−x_λ^k_p +2^p

∞ k=1

ρj x_λ^k_p

<2^pε^p+2^pρρ_j xλ_p

≤2^pε^p+2^pmj

(4.4)

(xλ ∈C = ∩j∈JAD(j,mj)), then if we let ε→0 we get ρρ_j(z)≤2mj, and finally z∈2C⊂p(E).

Theorem4.4. If Dist-complete, thenp(E)C isρC-complete.

Proof. Let(x_n^k)be aρC-Cauchy sequence; it is clearlyρC-bounded andτ-Cauchy, so(x_n^k)⊂sCfor somes∈N. Then byLemma 4.3, there is az=(z^k)∈2sC⊂p(E)C

such that the sequence (x_n^k) converges to z with respect to the topology τ. Note that AD is τ-closed so AD(j,m) is alsoτ-closed for every j∈J and m∈N; then C= ∩j∈JAD(j,mj)isτ-closed. Forε >0 there isN∈Nsuch that(x^kn)−(xm^k)∈εCfor everyn,m≥N, and sinceCisτ-closed(x^k_n)−(τ−lim(x_m^k))∈εCthen(x^k_n)−(z^k)∈ εCfor everyn≥N. This means that(x_n^k)converges to(z^k)with respect toρC.

Theorem4.5. IfEist-complete, thenp(E)isτ-complete.

Proof. The proof ofLemma 4.3can be repeated here to get theτ-completeness ofp(E).

Acknowledgments. C. Bosch and C. Gómez were partially supported by Aso- ciación Mexicana de Cultura A.C. T. Gilsdorf was partially supported by a Fulbright scholarship. R. Vera was supported by Coordinación de la Investigación Científica de la Universidad Michoacana.

References

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Soc. Mat. São Paulo8(1953), 1–79 (French).

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C. Bosch: Departamento Académico de Matemáticas, Itam, Rio Hondo #1, Col. Tizapán San Ángel, C.P.01000México D.F., Mexico

E-mail address:[email protected]

T. Gilsdorf: Department of Mathematics, University of North Dakota, Grand Forks, ND58202-8376, USA

E-mail address:[email protected]

C. Gómez: Departamento Académico de Matemáticas, Itam, Rio Hondo #1, Col. Tiza- pán San Ángel, C.P.01000México D.F., Mexico

E-mail address:[email protected]

R. Vera: Esc. de C. Físico-Matemáticas, Universidad Michoacana, Santiago Tapia No.

403Edif. B, Ciudad Universitaria,58060Morelia, Mich., Mexico E-mail address:[email protected]