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NON–ARCHIMEDEAN SEQUENTIAL SPACES AND THE FINEST LOCALLY CONVEX TOPOLOGY

WITH THE SAME COMPACTOID SETS

A. K. KATSARAS, C. PETALAS and T. VIDALIS

Abstract. For a non-Archimedean locally convex space (E, τ), the finest locally convex topology having the same asτ convergent sequences and the finest locally convex topology having the same asτcompactoid sets are studied.

Introduction

For a locally convex space E over the field of either the real numbers or the complex numbers, Webb investigated in [13] the finest locally convex topology on Ehaving the same convergent sequences as the original topology. Also, he studied the finest locally convex topology which has the same precompact sets.

In this paper we look at analogous problems for non-Archimedean spaces. For a non-Archimedean locally convex space (E, τ), we study the sequential locally convex topology τs which is the finest locally convex topology with the same as τ convergent sequences. Passing from τ to τs, we get that the category of non- Archimedean sequential locally convex spaces and continuous linear maps is a full coreflective subcategory of the category of all locally convex spaces. If τ is the weak topology ofc0, thenτscoincides with the norm topology ofc0which of course is not true in the classical case. For a zero dimensional topological spaceX and a non-Archimedean locally convex spaceE, we look at the problem of when is the space C(X, E), of all continuous E-valued functions onX, with the topology of either the pointwise convergence or the compact convergence, a sequential space.

In caseE is metrizable, it is shown thatC(X, E) is sequential iff it is bornological and this happens iffX isN-replete, whereNis the set of natural numbers.

For a non-Archimedean locally convex topology τ on E, we study the locally convex topology τc which coincides with the finest locally convex topology with the same as τ compactoid sets. The compactoid sets in non-Archimedean spaces are much more important than the precompact sets. As in the case of τs, we get that the category of all non-Archimedean locally convex spaces (E, τ) and continuous linear maps, for whichτ =τc, is a full coreflective subcategory of the

Received November 25, 1992; revised January 3, 1994.

1980Mathematics Subject Classification(1991Revision). Primary 46S10.

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category of all locally convex spaces. Ifτis the weak topology of`and if the field is non-spherically complete, it is shown that τs coincides with the finest locally convex topology which agrees with τ on norm bounded sets and with the finest polar topology having the same asτ compactoid sets.

1. Preliminaries

All vector spaces considered in this paper will be over a complete non-Archime- dean valued fieldKwhose valuation is non-trivial.

For a subsetSof a vector spaceEoverK, we will denote by co (S) the absolutely convex hull ofS. The edged hull Ae, of an absolutely convex subset A of E, is defined by: Ae=Aif the valuation of Kis discrete, andAe=∩{λA:|λ|>1}if the valuation ofKis dense (see [9]).

A subsetB, of a locally convex spaceEoverK, is called compactoid if, for each neighborhoodV of zero, there exists a finite subsetSofEsuch thatB⊂co (S)+V. For a non-Archimedean seminormponE, we will denote byEp the quotient space E/kerpequipped with the norm||[x]p||=p(x), where kerp={x:p(x) = 0}. By Eˆp we will denote the completion ofEp.

A seminormponE is called polar ifp= sup{|f|:f ∈E, |f| ≤p}, whereE is the algebraic dual space ofE. The locally convex spaceE is called polar if its topology is generated by a family of polar seminorms (see [9]).

Equivalently,Eis a polar space if it has a base at zero consisting of polar sets, i.e.

setsV withV =V00, whereV00is the bipolar ofV. For other notions refering to non-Archimedean locally convex spaces and for related results we refer to [9].

2. Sequential Spaces

Definition 2.1. A subsetV, of a locally convex spaceE, is called a sequential neighborhood of zero if every null sequence inElies eventually inV. The spaceEis called sequential if every convex sequential neighborhood of zero is a neighborhood of zero.

We have the following easily established

Lemma 2.2. LetV be an absolutely convex absorbing subset of a locally convex spaceE. Then,V is a sequential neighborhood of zero iff its Minkowski functional pV is sequentially continuous.

Proposition 2.3. For a locally convex space E, the following are equivalent:

(1)E is a sequential space.

(2)Every sequentially continuous seminorm on E is continuous.

(3) For every locally convex spaceF, every sequentially continuous linear map fromE toF is continuous.

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(4) For every Banach space F, every sequentially continuous linear map from E toF is continuous.

Proof. The equivalence of (1) and (2) follows from Lemma 2.2

(2)⇒(3) Letf:E→F be linear and sequentially continuous. IfV is a convex neighborhood of zero in F, then f1(V) is a convex sequential neighborhood of zero inE and hencef1(V) is a neighborhood of zero.

(4)⇒(2) Let pbe a sequentially continuous seminorm onE and consider the Banach space ˆEp. The canonical mappingπp: E→Eˆp is sequentially continuous and hence continuous, which implies thatpis continuous.

Let now (E, τ) be a locally convex space. The family of all convex sequential τ-neighborhoods of zero is a base at zero for a locally convex topology τs. The family of polar (with respect to the pairhE, Ei) sequential τ-neighborhoods of zero is a base at zero for a polar topologyτπs. We have the following

Proposition 2.4. 1) τs coincides with the coarsest sequential topology finer thanτ.

2) τ is sequential iffτ=τs.

3) τs is the finest locally convex topology on E having the same convergent sequences asτ.

4)Ifτ1is a locally convex topology onE such that everyτ-null sequence is also τ1-null, then τ1 is coarser then τs.

5) The topologiesτ andτs have the same bounded sets.

6) If F is a locally convex space and f:E → F a linear mapping, then f is τs-continuous iff it is sequentiallyτ-continuous.

7)τs is generated by the family of all non-Archimedean seminorms onE which are sequentiallyτ-continuous.

8) τπs is the finest of all polar topologies τ1 onE such that everyτ-convergent sequence is alsoτ1-convergent. Ifτ is polar, thenτ ≤τπsand the topologiesτ and τπs have the same convergent sequences.

9) τπs≤τs.

10) τπs is generated by the family of all sequentially τ-continuous polar semi- norms onE.

11)τπs is the largest of all polar topologies which are coarser thanτs.

Proposition 2.5. Let (E, τ) and (F, τ1) be locally convex spaces. If a linear map

f: (E, τ)→(F, τ1)

is continuous, thenf is also (τs, τ1s)-continuous and (τπs,(τ1)sπ)-continuous.

Using the preceding Proposition we get that the category of non-Archimedean sequential locally convex spaces and continuous linear maps is a full coreflective subcategory of the category of all locally convex spaces.

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Corollary 2.6. If(E, τ) =Q

αI(Eα, τα), then τs≥Y

αI

α)s and τπs≥Y

αI

α)sπ.

Proposition 2.7. Let(E, τ) =Qn

k=1(Ek, τk). Then τs=

Yn k=1

k)s and τπs= Yn k=1

k)sπ.

Proof. Let τ0 = Qn

k=1k)s. By the preceding Corollary, we have τ0 ≤ τs. On the other hand, let V be a convex sequential τ-neighborhood of zero in E.

If jk: Ek → E is the canonical injection, then Vk = jk1(V) is a sequential τk- neighborhood of zero inEk. It follows thatW =Qn

k=1Vk is aτ0-neighborhood of zero withW ⊂V, which proves thatτs≤τ0.

The proof for the case ofτπs is analogous.

Recall that a locally convex space E is called polarly bornological (see [9]) if every subset ofE, which is polar with respect to the pairhE, Eiand which absorbs bounded sets is a neighborhood of zero. Equivalently,E is polarly bornological if every polar seminorm onE, which is bounded on bounded sets, is continuous.

Proposition 2.8. Let(E, τ) be a locally convex space.

1) If(E, τ)is bornological, thenτ=τs. 2) If(E, τ)is polarly bornological, then τπs≤τ.

Proof. 1) It follows from the fact thatτ andτshave the same bounded sets.

2) Let p be a sequentially continuous polar seminorm. Then, p is bounded on bounded sets. In fact, let B be a bounded set with supxBp(x) = ∞. Let

|λ| > 1 and choose a sequence (xn) in B with p(xn) >|λ|n. Now, λnxn → 0 butp(λnxn)≥1 for alln, a contradiction. SinceE is polarly bornological, pis continuous and the result follows from Proposition 2.4.

Proposition 2.9. If E is finite dimensional, then it is sequential.

Proof. IfE is Hausdorff, thenE is topologically isomorphic toKn, wheren= dim(E), and soE is sequential. IfE is not Hausdorff, let F ={0}. Since E/F is Hausdorff and finite-dimensional, its topology is given by some norm k · k. If π:E → E/F is the quotient map, then it is easy to see that the topology of E is given by the seminormp(x) =kπ(x)k. ThusE is seminormable and hence

sequential.

We have also the following easily established

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Proposition 2.10. Let{Eα : α∈I}be a family of locally convex spaces, E a vector space and, for each α∈ I, fα:Eα →E a linear mapping. If each Eα

is sequential and ifE is equipped with the finest locally convex topology for which each fα is continuous, thenE is sequential.

Corollary 2.11. Quotient spaces and direct sums of sequential locally convex spaces are sequential.

The result about direct sums of sequential spaces also follows using the general theory of coreflective subcategories.

For a locally convex space (E, τ), we will denote byEsthe space of all sequen- tiallyτ-continuous linear functionals onE. ClearlyEs= (E, τs)0.

Definition 2.12. A subset B of E is called sequentially τ-equicontinuous if xnτ 0 in E implies that f(xn) → 0 uniformly for f ∈ B, i.e.

limn→∞supfB|f(xn)| = 0. Clearly every sequentially τ-equicontinuous subset ofE is contained inEsandEs is the union of all such subsets ofE.

Lemma 2.13. If B ⊂ E is sequentially τ-equicontinuous, then its bipolar B, with respect to the pairhE, Ei, is also sequentiallyτ-equicontinuous.

Proof. It follows from the fact that for eachx∈E we have

fsupB

|f(x)|= sup

fB

|f(x)|.

In the following Proposition, we will denote by b(Es, E) the strong topology onEs.

Proposition 2.14. Ifτ is polar, then every sequentiallyτ-equicontinuous sub- setH of Esis b(Es, E)-bounded.

Proof. Assume thatHis not strongly bounded and letAbe aσ(E, Es)-bounded subset ofE such that

xA, fsupH

|f(x)|=∞.

Since E0 ⊂Es, the set A isσ(E, E0)-bounded and hence it isτ-bounded sinceτ is polar.

Let |λ| >1 and choose a sequence (xn) in A and a sequence (fn) in H such that |fn(xn)| ≥ |λ|n for all n. Since A is bounded, we have that λnxnτ 0.

Moreover, |fnnxn)| ≥ 1 which contradict the fact that H is sequentially τ-

equicontinuous.

Since, for eachf ∈Es, the seminorm pf(x) =|f(x)| is polar and sequentially continuous, it is clear thatEs= (E, τπs)0.

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Proposition 2.15. τπs coincides with the topology of uniform convergence on the sequentiallyτ-equicontinuous subsets ofEs.

Proof. It is easy to see that a subset ofEsisτπs-equicontinuous iff it is sequen- tially τ-equicontinuous. Now the result follows form the fact that τπs is a polar

topology.

Notation. For a locally convex space (E, τ), we will denote by Eb the space of all bounded linear functionals on E, i.e. the space of all f ∈ E which are bounded onτ-bounded sets. Byτn we will denote the topology onEbof uniform convergence on theτ-null sequences inE.

Proposition 2.16. For an absolutely convex subset H of Es, the following assertions are equivalent:

(1)H is sequentially τ-equicontinuous.

(2)H isτn-compactoid.

Proof. (1)⇒(2). Letpbe the Minkowski functional of the polarH0ofH inE.

We have that

p(x) = sup

fH

|f(x)| (x∈E).

LetA={xn:n∈N}, where (xn) is aτ-null sequence inE. Let 0<|µ|<1 and letε >0 with 4ε≤ |µ|. There exists an indexn0 such thatp(xn)< εifn > n0. We choose a basis{z1, . . . , zk}for F = [x1, . . . , xn0] which is 12-orthogonal with respect to the seminormp. We may assume thatp(z1)≥p(z2)≥ · · · ≥p(zk).

Letm≤kbe such thatp(zi)>0 ifi≤mandp(zi) = 0 ifi > m. We may assume that|µ| ≤p(zi)<1 fori≤m. There areg1, . . . , gminG0, whereG= [z1, . . . , zm], withgi(zj) = 0 ifi6=j andgi(zi) = 1. Forx=Pm

i=1λizi∈G, we have p(x)≥ 1

2 sup

1im

i|p(zi)≥ sup

1im

|µλi| 2 . Thus,

|gi(x)|=|λi| ≤ 2

|µ|p(x).

Sincepis a polar seminorm, there exists a continuous extension gi ofgi to all of E such that

|gi(y)| ≤ 4

|µ|p(y) for ally∈E. Note thatgi∈(E, τπs)0=Es. Let nowf ∈H and set h=Pm

i=1f(zi)gi. Sincef ∈H, we have|f| ≤pand so

|f(zi)| ≤p(zi)≤1, which implies thath∈co (g1, . . . , gm). Moreover,h=f onG.

Form < i≤k, we have

|gi(zi)| ≤ 4

|µ|p(zi) = 0,

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which implies thath=f onF. Finally, forn > n0, we have

|f(xn)| ≤p(xn)<1, |gi(xn)| ≤ 4

|µ|p(xn)≤ 4ε

|µ| ≤1.

Therefore,|(f−h)(xn)| ≤1 for allnand sof−h∈A0, whereA0 is the polar of AinEb. Thus

H ⊂co (g1, . . . , gm) +A0, which proves thatH isτn-compactoid.

(2)⇒(1). Let (xn) be aτ-null sequence inE and setA={xn:n∈N}. Since the polarA0 ofAinEb is aτn-neighborhood of zero, givenµ6= 0 inKthere are g1, . . . , gm in the linear hull [H]⊂Es ofH such that

H ⊂co (g1, . . . , gm) +µA0.

Letn0be such that|gk(xn)| ≤ |µ|, fork= 1, . . . , m, ifn≥n0. Now

fsupH

|f(xn)| ≤ |µ|

for all n ≥n0. In fact, let f ∈ H. There exist g ∈ co (g1, . . . , gm) and h ∈ A0 such thatf =g+µh, which implies that, forn≥n0, we have|f(xn)| ≤ |µ|since

|g(xn)| ≤ |µ|. This proves thatH is sequentiallyτ-equicontinuous.

3. The Topologyτc

In this section we will study the finest locally convex topology onEhaving the same compactoid sets as a given locally convex topology.

Proposition 3.1. Let τ1, τ2 be locally convex topologies on E such that ev- ery τ1-compactoid is also τ2-compactoid. Then, every τ1-bounded set is also τ2- bounded.

Proof. Let A be a subset ofE which is τ1-bounded but not τ2-bounded. We may assume thatAis absolutely convex. SinceAis notτ2-bounded, givenµ∈K, with|µ|>1, there exist a convexτ2-neighborhoodV of zero and a sequence (xn) in A with xn ∈/ µ2nV. The sequence (yn), yn = µnxn, is τ1-null and hence τ1-compactoid, which implies that (yn) is τ2-compactoid. Therefore, (yn) is τ2- bounded and soµnynτ2 0, which is a contradiction sinceµnyn∈/V.

Let now τ be a locally convex topology on E and let Bτ be the family of all convex absorbing subsets V of E with the following property: For each τ- compactoid subset A of E there exists a finite subset S of E such that A ⊂ co (S) +V. Clearly every convexτ-neighborhood of zero is in Bτ and, for each V ∈Bτ and eachµ /∈0, we haveµV ∈Bτ. IfV1, V2 are inBτ, thenV =V1∩V2

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is also inBτ. In fact, let Abe an absolutely convexτ-compactoid and let|λ|>1.

There exists a finite subsetS={x1, . . . , xn}ofE such that A⊂co (S) +λ2V1.

By [3, Lemma 1.2], there exists a finite subsetS1={y1, . . . , yn}of λAsuch that A⊂co (S1) +λ1V1.

Since the set B = [A+ co (S1)]∩(λ1V1) is a τ-compactoid, using again [3, Lemma 1.2], we can find a finite subsetS2ofλB⊂V1 such that

B⊂co (S2) +V2. Now

A⊂co (S1∪S2) +V1∩V2.

In fact, let x ∈ A. There existsz1 ∈ co (S1) such that x−z1 ∈ λ1V1. Since x−z1 ∈ B, there exists z2 ∈ co (S2) ⊂ V1 such that x−z1−z2 ∈ V2. Since x−z1∈B⊂λ1V1⊂V1, we have thatx−z1−z2∈V1∩V2andz1+z2∈co (S1∪S2), which completes the proof of our claim. This proves thatV1∩V2∈Bτ. It follows from the above that Bτ is a base at zero for a locally convex topology τc finer

thanτ.

Proposition 3.2. (1)τc is the finest locally convex topology on E having the same compactoid sets asτ.

(2)If τ1 is a locally convex topology onE such that everyτ-compactoid is also τ1-compactoid, then τ1≤τc.

(3)The topologies τ andτc have the same bounded sets.

(4)If (E, τ) is bornological, thenτ=τc.

(5) If F is a locally convex space and f:E →F a linear mapping, then f is τc-continuous iff it maps τ-compactoid sets into compactoid sets in F.

(6) τc = τ iff, for any locally convex space F, any linear map f: E → F mappingτ-compactoid sets into compactoid sets isτ-continuous.

(7)τc=τiff, for any Banach spaceF, any linear functionf: E→F, mapping τ-compactoid sets into compactoid sets, isτ-continuous.

Proof. (1) and (2) follows easily from the definitions.

(3) It follows from (1) and Proposition 3.1.

(4) It follows from (3) sinceτ ≤τc.

(5) Necessity follows from (1) since images of compactoid sets, under continu- ous linear mappings, are compactoid. For the sufficiency, let the linear function f:E → F map τ-compactoid sets into compactoid sets and let V be a convex neighborhood of zero in F. Let A be an absolutely convex τ-compactoid in E

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and let |λ|>1. Since f(A) is compactoid inF, there exists a finite subsetT of λf(A) such thatf(A)⊂co (T) +V. IfS is a finite subset of λAwithT =f(S), then A ⊂ co (S) +f1(V). This proves thatf1(V) ∈ Bτ and so f1(V) is a τc-neighborhood of zero.

(6) Necessity follows from (5). To prove the sufficiency of the condition, it suffices to takeF = (E, τc) and consider the identity map fromE toF.

(7) Suppose that, for any Banach space F, any linear function from E to F, which mapsτ-compactoid sets into compactoid sets, is continuous. Letpbe aτc- continuous non-Archimedean seminorm onE and consider the Banach spaceG= Eˆp. The canonical mappingϕp:E→Gmapsτ-compactoid sets into compactoid sets, and soϕp is continuous, which implies thatpisτ-continuous.

Notation 3.3. We will denote byτπc the finest of all polar topologies τ1 onE such that everyτ-compactoid set is alsoτ1-compactoid.

We have the following easily established

Lemma 3.4. a)τπc is the finest polar topology on E coarser thanτc.

b) If τ is polar, then τ ≤τπc and the two topologies τ and τπc have the same compactoid sets and the same bounded sets.

c) Everyτ-bounded set is τπc-bounded.

Let us recall next the notion of the Kolmogorov diameters of a bounded set.

Ifpis a non-Archimedean seminorm on E and Aap-bounded set, then for each non-negative integern then-th Kolmogorov diameter δn,p(A) of A, with respect to p, is the infimum of all|µ|, µ∈K, for which there exists a subspaceF ofE, with dimF≤n, such that

A⊂F+µBp(0,1), whereBp(0,1) ={x∈E:p(x)≤1}(see [8]).

By [8], a subset A of E is τ-compactoid iff limn→∞δn,p(A) = 0 for each τ- continuous seminormponE.

Lemma 3.5. A non-Archimedean seminorm p on E is τ-bounded, i.e. it is bounded on bounded sets, iffpis bounded on τ-compactoid sets.

Proof. Assume that there exists aτ-bounded setAsuch that supxAp(x) =∞. Given |λ| > 1, there exists a sequence (xn) in A with p(xn)> |λ|2n. Now, the sequence (λnxn) isτ-null and hence τ-compactoid but supnp(λnxn) =∞. Proposition 3.6. Let Pτ be the family of all τ-bounded non-Archimedean seminorms p on E such that limn→∞δn,p(A) = 0 for each τ-compactoid set A.

Then:

a) If p∈ Pτ and ifq is a non-Archimedean seminorm on E with q ≤p, then q∈ Pτ.

b) Ifp1, p2∈ Pτ, thenp1+p2 andp= max{p1, p2}are also in Pτ.

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c) Ifp∈ Pτ, then|µ|p∈ Pτ for eachµ∈K.

Proof. Letp∈ Pτ and letµ6= 0. Given aτ-compactoid setA, there exists ann such thatδn,p(µA)<|µ| sinceµAis τ-compactoid. By [8, Proposition 3.2], there arex1, . . . , xn inE such that

µA⊂co (x1, . . . , xn) +µBp(0,1) and so

A⊂co (µ1x1, . . . , µ1xn) +Bp(0,1). This proves thatBp(0,1) is inBτ.

Let nowp1, p2 ∈ Pτ and µ6= 0. Chooseλ∈Kwith 0<|λ|<|µ|/2. Since both λBp1(0,1) andλBp2(0,1) are inBτ, the same is true for the set

V = [λBp1(0,1)]∩[λBp2(0,1)] =λ[Bp1(0,1)∩Bp2(0,1)].

Ifp=p1+p2, thenV ⊂µBp(0,1) and soBp(0,1)∈Bτ. It follows that there are y1, . . . , yminE such that

A⊂co (y1, . . . , ym) +µBp(0,1)

and soδm,p(A)≤ |µ|. Thus, forn≥m, we haveδn,p(A)≤ |µ|, which proves that δn,p(A)→0 and sop∈ Pτ. The proofs of the other assertions in the Proposition

follow easily from the definitions.

Proposition 3.7. (1)A non-Archimedean seminorm ponE isτc-continuous iff it belongs toPτ.

(2)The family of all polar members of Pτ generates the topology τπc.

Proof. (1) If p ∈ Pτ, then, as we have seen in the proof of the preceding Proposition,Bp(0,1) belongs toBτ and sopisτc-continuous. Conversely, letpbe τc-continuous. IfAisτ-compactoid, thenAisτc-compactoid and soδn,p(A)→0, which proves thatp∈ Pτ.

(2) The proof is analogous to that of (1).

We have the following easily established

Proposition 3.8. If a linear mapf: (E, τ)→(F, τ1)is continuous, thenf is (τc, τ1c)-continuous and (τπc,(τ1)cπ)-continuous.

In view of the preceding Proposition, we get that the category of all locally convex spaces (E, τ), withτc =τ, and continuous linear maps is a full coreflective subcategory of the category of all locally convex spaces.

Corollary 3.9. If (E, τ) = Q

αI(Eα, τα), then τc ≥ Q

αIα)c and τπc ≥ Q

αIα)cπ.

The proof of the following Proposition is analogous to that of Proposition 2.7.

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Proposition 3.10. If (E, τ) = Qn

k=1(Ek, τk), then τc = Qn

k=1τkc and τπc = Qn

k=1k)cπ.

Proposition 3.11. (E, τc)0 = (E, τπc)0 =Eb.

Proof. IfAisτ-compactoid, thenAisτ-bounded and soAisσ(E, Eb)-bounded, which implies A is σ(E, Eb)-compactoid. Since σ(E, Eb) is a polar topology, we have thatσ(E, Eb)≤τπc and so

Eb= (E, σ(E, Eb))0≤(E, τπc)0.

On the other hand, let f ∈ (E, τc)0 and let A be a τ-bounded set. Then, A is τc-bounded and so f is bounded onA which proves thatf ∈Eb. We will need the following Proposition which is analogous to the Grothendieck’s interchange Theorem. We will say that a familyM of subsets of a vector space G is directed if given M1, M2 ∈ M there exists M3 ∈ M containing both M1

andM2.

Proposition 3.12. Let hE, Fi be a dual pair of vector spaces over Kand let M(resp.N) be a directed family ofσ(E, F)-bounded (resp.σ(F, E)-bounded) sub- sets ofE (resp. F) covering E (resp. F). On E we consider the topology τN of uniform convergence on the members ofN and on F the topologyτM of uniform convergence on the members ofM. Then, the following statements are equivalent:

(1)Each member of MisτN-compactoid.

(2)Each member of N is τM-compactoid.

Proof. (1)⇒(2). Without loss of generality, we may assume that all members ofMandN are absolutely convex. LetH ∈ N,M ∈ Mandγ6= 0. Sinceγ1M isτN-compactoid, given|λ|>1 there arex1, . . . , xn inE such that

γ1M ⊂co (x1, . . . , xn) +λ1H0. The set

D={(f(x1), . . . , f(xn)) :f ∈H}

is bounded in Kn. Let µ ∈ K be such that |f(xk)| ≤ |µ| for allf ∈ H and for k = 1,2, . . . , n. If z(k) = (0,0, . . . , µ, . . . ,0), where µ is in the k-position, then D ⊂co (z(1), . . . , z(n)). By [3, Lemma 1.2], there are f1, . . . , fn in H such that D⊂λco (c1, . . . , cn), whereck= (fk(x1), . . . , fk(xn)). Let nowf ∈H. There are γ1, . . . , γn inK,|γk| ≤1 such that

f(xi) =λ Xn k=1

γkfk(xi), i= 1, . . . , n .

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Leth=f −λPn

k=1γkfk. Ifx∈M, then γ1x=

Xn i=1

λixi+z, |λi| ≤1, z∈λ1H0. We have

h Xn

i=1

λixi

!

= Xn i=1

λi f(xi)−λ Xn k=1

γkfk(xi)

!

= 0. Thus,

|h(γ1x)|=

f(z)−λ Xn k=1

γkfk(z) ≤1

sincef, fk∈H andz∈λ1H0. This proves thath∈γM0. Therefore, H ⊂λco (f1, . . . , fn) +γM0,

which proves thatH isτM-compactoid.

(2)⇒(1). The proof is analogous.

Letτ0denote the topology onEb of uniform convergence on theτ-compactoid subsets ofE. Byτ00 we will denote the topology onE of uniform convergence on theτ0-compactoid subsets ofEb.

Proposition 3.13. If a set B ⊂ Eb is τ0-compactoid, then its bipolar B00, with respect to the pair

E, Eb

, is also τ0-compactoid.

Proof. LetA⊂E beτ-compactoid and let A0 be its polar inEb. Let|λ|>1 and letf1, . . . , fn inEb be such that

B⊂co (f1, . . . , fn) +λ1A0.

By an argument similar to the one used by Schikhof in [9, Corollary 5.8], we get that

B00⊂[co (f1, . . . , fn) +λ1A0]00⊂[co (f1, . . . , fn) +λ1A0]e

⊂co (λf1, . . . , λfn) +A0,

which proves thatB00isτ0-compactoid.

Lemma 3.14. If τ is polar, thenτ ≤τ00.

Proof. Consider the dual pairhE, E0i, E0= (E, τ)0, and takeM the family of allτ-compactoid subsets ofE andN the family of allτ-equicontinuous subsets of E0. Since τ is polar, we have thatτ =τN. Using Proposition 3.12, we get that eachH ∈ N is τM-compactoid and so H is τ0-compactoid. If nowV is a polar τ-neighborhood of zero, we have thatV =V00 is aτ00-neighborhood of zero, and

this proves thatτ≤τ00.

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Proposition 3.15. For every locally convex space(E, τ), we have τ00πc. Proof. LetMbe the family of allτ-compactoid subsets ofEandN the family of allτ0-compactoid subsets ofEb. SinceτM0 andτN00, it follows from Proposition 3.12 that everyτ-compactoid set isτ00-compactoid and soτ00 ≤τc. Since τ00 is a polar topology, we have that τ00 ≤ τπc. On the other hand, if τ1πc, then everyτ-compactoid set isτ1-compactoid and so everyτ-bounded set isτ1-bounded, which implies that G= (E, τ1)b ⊂Eb andτ0|G≤τ10. IfH ⊂Gis τ10-compactoid, thenH isτ0-compactoid, and this implies thatτ100≤τ00. Sinceτ1

is polar, we have (by the preceding Proposition)τ1≤τ100≤τ00and soτ00πc. Proposition 3.16. τ00 = τπc is the finest of all polar topologies on E which agree withσ(E, Eb)on τ-compactoid sets.

Proof. The topologyτ00is polar and (E, τ00)0=Eb. IfA⊂Eisτ-compactoid, thenA isτ00-compactoid and soτ00 =σ(E, Eb) onAby [9, Theorem 5.12]. On the other hand, let τ1 be a polar topology on E agreeing with σ(E, Eb) on τ- compactoid sets. Let A be an absolutely convex τ-compactoid set. Then, A is σ(E, Eb)-bounded and hence A is σ(E, Eb)-compactoid. Since τ1 =σ(E, Eb) on A, it follows that A is τ1-compactoid by [10, Proposition 4.5]. Thus, every τ-compactoid set isτ1-compactoid, which implies thatτ1≤τ00sinceτ1 is polar.

Corollary 3.17. An absolutely convex subsetV of E is a τπc-neighborhood of zero iff for eachτ-compactoid setA there are f1, . . . , fn inEb such that

\n k=1

{x∈E :|fk(x)| ≤1} ∩A⊂V .

Corollary 3.18. Let τ be a polar topology and assume that Eb =E0. Then, τπc is the finest of all polar locally convex topologies on E which agree with τ on τ-compactoid sets.

Proof. It follows from Proposition 3.16 sinceτ =σ(E, E0) onτ-compactoid sets

by [9, Theorem 5.12].

Proposition 3.19. For a locally convex space (E, τ), the following are equiv- alent:

(1)τπs≤τπc.

(2)Everyτ-compactoid set is τπs-compactoid.

(3)Every sequentially τ-equicontinuous subset ofEb isτ0-compactoid.

(4)The topologies τπs andσ(E, Es) coincide onτ-compactoid sets.

(5)The topologiesτ0andσ(Eb, E)coincide on every sequentiallyτ-equicontinu- ous subset ofEb.

Proof. (1) ⇒ (2) It follows from the fact that every τ-compactoid set is τπc- compactoid.

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(2)⇒(1) It is obvious.

(1) ⇒ (3) Let H ⊂ Eb be sequentiallyτ-equicontinuous. Then, H0 is a τπs- neighborhood of zero and so H0 is a τπc-neighborhood of zero. Therefore, there exists aτ0-compactoid subsetBofEbsuch thatB0⊂H0. It follows thatH ⊂B00 andB00 isτ0-compactoid by Proposition 3.13.

(3)⇒(1) It follows from Propositions 2.15 and 3.15.

(2) ⇒ (4) Since Es = (E, τπs)0, the topologies τπs and σ(E, Es) coincide on τπs-compactoid sets (by [9, Theorem 5.12]) and so they coincide onτ-compactoid sets.

(4)⇒(2) LetA be an absolutely convexτ-compactoid. Then,Aisτ-bounded and soAisσ(E, Es)-bounded, which implies thatA isσ(E, Es)-compactoid. By [10, Proposition 4.5],Ais τπs-compactoid.

(3) ⇒ (5) Let H ⊂ Eb be sequentially τ-equicontinuous. Then, H is τ0- compactoid. Consider the topologies τ0 and σ(Eb, E). The topology τ0 is finer thanσ(Eb, E) and it has a base at zero consisting of absolutely convexσ(Eb, E)- closed sets. By [10, Theorem 1.4],τ0=σ(Eb, E) onH.

(5)⇒(3) LetH be a sequentiallyτ-equicontinuous subset ofEb. Without loss of generality, we may assume thatH is absolutely convex. SinceH isσ(Eb, E)- bounded, it isσ(Eb, E)-compactoid and so H is τ0-compactoid by [10, Proposi-

tion 4.5].

Proposition 3.20. For every locally convex space (E, τ), the following are equivalent:

(1)τπc≤τπs. (2)Es=Eb.

(3)τπscoincides with the topology of uniform convergence on theτn-compactoid subsets ofEb.

(4)τπs≥(τπc)sπ.

(5)τπsis finer than the topology of uniform convergence on theτn-null sequences inEb.

Proof. (1)⇒(2)

Eb= (E, τπc)0 ⊂(E, τπs)0 =Es⊂Eb. (2)⇒(3) It follows from Proposition 2.15 and 2.16.

(3)⇒(1) Letτ1be the topology of uniform convergence on theτn-compactoid subsets ofEb. Since everyτ-null sequence isτ-compactoid, we have thatτn ≤τ0 and hence everyτ0-compactoid is alsoτn-compactoid. Therefore,τ1≥τ00πc.

(1)⇒(4) It is obvious.

(4)⇒(1) Sinceτπc is polar, we have

τπc≤(τπc)sπ≤τπs.

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(3)⇒(5) It follows from the fact that everyτn-null sequence isτn-compactoid.

(5)⇒(2) Letτ2be the topology of uniform convergence on theτn-null sequences inEb. Sinceτ2is finer thanσ(E, Eb), we have thatEb⊂(E, τ2)0 and so

Es⊂Eb⊂(E, τ2)0⊂(E, τπs)0 =Es.

Note. Ifτ is polar, then τ≤τπc and soτπs≤(τπc)sπ. Hence, in case of polarτ, in the preceding Proposition (4) may be replaced byτπs= (τπc)sπ.

Proposition 3.21. If(E, τ)is a polar space which is polarly bornological, then τ=τπsπc.

Proof. Letf ∈Eb. The seminormp(x) =|f(x)|is polar and it is bounded on τ-bounded sets and so pis τ-continuous, since (E, τ) is polarly bornological. It follows that everyf ∈Eb isτ-continuous and so Eb =Es=E0. Sinceτ is polar, we haveτ ≤τπc. Thus, by Propositions 2.8 and 3.20, we haveτ ≤τπc≤τπs≤τ.

Examples. Taking asτeither the topologyσ(`, c0) or the topologyσ(c0, `), we will look at the topologiesτs, τπscπc.

Every element y of ` defines a continuous linear functional fy on c0 by fy(x) =hx, yi=P

n=1xnyn. Moreover,kfyk=kyk. Using the principle of uni- form boundedness, we get that a subset of` is norm-bounded iff it isσ(`, c0)- bounded. Also, a subset of c0 is norm-bounded iff it is σ(c0, `)-bounded by [9, Corollary 7.7].

We will need the following

Proposition 3.22. Let τ = σ(`, c0) and let τ1 be the finest locally convex topology on`agreeing withτ on norm-bounded (equivalently onτ-bounded) sets.

Then:

1) The topology τ1 is polar.

2)Ifτ2is the metrizable locally convex topology on`generated by the countable family of seminorms{pn :n∈N},pn(x) =|xn|, thenτ2coincides withτon norm- bounded sets and hence τ|A is metrizable on eachτ-bounded set A.

Proof. 1) LetBn ={x∈`:kxk ≤n}. By [5, Theorem 5.2],τ1 has a base at zero consisting of setsW of the form

(*) W =W0

\ n=1

(Bn+Wn)

! ,

where each Wn is a τ-neighborhood of zero. We may assume that, for eachn, there exists a finite subsetSn of c0 such thatWn =Sn0, whereSn0 is the polar of Sn in`. If F = (`, τ1), then eachWn is a polar set in F. Note thatτ ≤τ1

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and soc0⊂F0. SinceBn isτ-bounded, it isτ-compactoid and hence, for|λ|>1, there exists a finite subsetAn ofBn such that

Bn⊂λc0(An) +Wn.

Taking bipolars with respect to the pairhF, F0i and using [9, Corollary 5.8], we get

(Bn+Wn)00⊂(λc0(An) +Wn)00= (λc0(An) +Wn)e

⊂λ(λc0(An) +Wn ⊂λ2(c0(An) +Wn)⊂λ2(Bn+Wn) and soW00⊂λ2W. This proves thatτ1has a base at zero consisting of polar sets and so it is a polar topology.

2) Clearly τ2 ≤τ. To show that τ2 =τ on norm-bounded sets, let (xα) be a norm-bounded net inEwithxα τ2 0. Letε >0 and letd≥supαkxαk. Lety∈c0

and chooseε1 >0 withε1(d+kyk)< ε. Letmbe such that|yn|< ε1 ifn > m.

Ifα0 is such that|xαk|< ε1 for allα≥α0and allk= 1,2, . . . , m, then forα≥α0

we have

| hxα, yi |=

X

n

xαnyn

≤ε .

The proof of the next Proposition is similar to the one of Proposition 3.22.

Proposition 3.23. 1) On each norm-bounded subset ofc0, the weak topology τ =σ(c0, `) coincides with the metrizable locally convex topology generated by the countable family of seminorms{pn :n∈N},pn(x) =|xn|.

2) If τ3 is the finest locally convex topology on c0 agreeing with τ on norm- bounded sets. then τ3 is polar.

Example I. LetE = ` and τ =σ(`, c0). If τ1 is as in Proposition3.22, then:

1) τsπs1 and each of these topologies is strictly coarser than the norm topologyτk·k and strictly finer thanτ.

2) IfKis not spherically complete, then τ1πssπc.

Proof. We show first that τ =τs on each norm-bounded setA. To show that τs|A≤τ|A, it suffices to prove that the identity map

I: (A, τ|A)→(A, τs|A)

is continuous. Since τ|A is metrizable, it suffices to show that I is sequentially continuous, which is clearly true. Thusτ|A=τs|Aand soτs≤τ1by the definition of τ1. To prove that τ1 ≤τs, it is sufficient to show that every τ-null sequence

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is τ1-null. So, let (yn) be a τ-null sequence. Then (yn) isτ-bounded and hence norm-bounded. Sinceτ =τ1on norm-bounded sets, we have that (yn) is aτ1-null sequence. Thusτ1s. In view of Proposition 3.2, the topologyτsis polar and so τsπs. The topologyτ1 is coarser than the norm topology. In fact, letV be an absolutely convex subset ofEwhich is not a norm-neighborhood of zero. There exists a sequence (yn) in E, kynk ≤ 1

n, yn ∈/V. Since τ1 =τ on norm-bounded sets and since yn τ→ 0, it follows that yn τ1 0 and this implies thatV is not a τ1-neighborhood of zero. So,τ1≤τk·k. Butτ16=τk·k. In fact, leten∈`,enk = 0 ifk6=nand enn = 1. The sequence (en) is τ-null and hence τ1-null, since (en) is norm-bounded, but (en) is not a norm-null sequence.

Finally,τ1is strictly finer thanτ. In fact, let 0<|λ|<1. The sequence (λnen) is a sequentiallyτ-equicontinuous subset of Eb. IfH ={λnen :n∈N}, then the polarH0 ofH inE is aτπs-neighborhood of zero by Proposition 2.15. ButH0is not aτ-neighborhood of zero. In fact, assume that there exists a finite subset S ofc0 such thatS0⊂H0. Using [11, Corollary 1.2], we get that the set [c0(S)]eis σ(c0, l)-closed. Taking bipolars with respect to the pairhc0, li, and using [9, Proposition 4.10], we get

H ⊂H00⊂S00= [c0(S)]00

=

c0(S)σ(c0, l) e

= [c0(S)]e⊂λc0(S)

for|λ|>1, which cannot hold sinceH is linearly independent. This proves that τ is strictly coarser thanτπs1.

2) Assume thatK is not spherically complete. Then (`,k · k)0 =c0. Since a subset of` is norm-bounded iff it isτ-bounded, it follows that

Es=Eb =c0= (`,k · k)0 = (E, τ)0.

Since a subsetA ofE isτ-compactoid iff it isτ-bounded and this is true iff Ais norm-bounded, it follows from Corollary 3.18 that τπc1. Also since a subset A of E is τ-compactoid iff it is norm-bounded, it follows that the topology τ0 on Eb =c0 is the norm topology ofc0. As it is well known, a subset H of c0 is norm-compactoid iff there existsz∈c0such that

H ⊂Hz={y∈c0:|yn| ≤ |zn| for alln}.

Ifpzis defined on`bypz(x) = supn|znxn|, then the polar ofHzin`coincides with the set

{x∈`:pz(x)≤1}.

Thusτ00πc is generated by the family of seminorms{pz:z∈c0}.

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Example II. LetE=c0 andτ=σ(c0, `). Then 1) τsπsk·k.

2)τcπc and this topology is strictly coarser than the norm topologyτk·kofc0. Proof. 1) It is well known that a sequence in c0 is norm-convergent iff it is weakly convergent (see [12, p. 158]). It follows from this that the norm topology τk·k onc0is coarser thanτπssince the norm topology is polar. On the other hand, every norm-convergent sequence isτ-convergent and henceτs-convergent, which implies thatτs≤τk·k. Thereforeτsπsk·k.

2) There are norm-bounded subsets ofc0 which are not norm-compactoid. If A is such a set, then A is τ-compactoid (since it is τ-bounded) and hence τc- compactoid. This implies that τc 6= τk·k. If V is an absolutely convex subset of E which is not a norm-neighborhood of zero and if 0 < |λ| < 1, then there exists a sequence (yn) in E with yn ∈/ V and kynk ≤ |λ|n. Now the sequence (zn) = (λnyn) is τ-bounded and hence τc-bounded. Sincezn ∈/λnV for alln, it follows thatV is not a τc-neighborhood of zero. This proves that τc is strictly coarser that the norm topology. To prove thatτcπc, we consider the topology τ3defined in Proposition 3.23. LetAbe aτ-bounded set and consider the identity map

I: (A, τ|A)→(A, τc|A).

If (yn) is a sequence inAwithyn τ→x, thenyn→xin the norm topology, which implies thatyn τc xsinceτcis coarser thanτk·k. ThusIis sequentially continuous and henceI is continuous sinceτ|A is metrizable by Proposition 3.23. It follows thatτc|A =τ|A, for eachτ-bounded set A, and soτc ≤τ3. It is also clear that Eb=c00=` since theτ-bounded subsets ofE coincide with the norm-bounded sets. ThusEb= (E, τ)0. Sinceτandτ3are polar topologies, we have thatτ3πc

and soτcπc3.

Remark 3.24. Let F be a subspace of a locally convex space (E, τ) and let τ1=τ|F. It is easy to see thatτc|F ≤τ1cπc|F ≤(τ1)cπs|F ≤τ1sπs|F ≤(τ1)sπ. The following is an example whereτπc|F 6= (τ1)cπs|F 6=τ1sπs|F 6= (τ1)sπ.

Example. Assume thatKis not spherically complete and takeE =`, τ = σ(`, c0). As we have seenτπcsπs. LetF =c0, τ1=τ|F. Since a subset ofF isτ1-bounded iff it is norm-bounded, it follows that (F, τ1)b=c00 =`. Let z∈`\c0. The set

V ={x∈c0:| hx, zi | ≤1}

is aτ100= (τ1)cπ neighborhood of zero. ButV is not a neighborhood of zero with respect to the topologyτ2πc|F. In fact, if V were a τ2-neighborhood of zero, then there would existy∈c0 such that

(*) W ={x∈c0:py(x)≤1} ⊂V .

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It follows easily from (*) that|zn| ≤ |yn|, for alln, and soz∈c0, a contradiction.

Thus τ2 is strictly coarser than (τ1)cπ. Since V is clearly a convex sequentialτ1- neighborhood of zero, it follows thatτs|Fis strictly coarser thanτ1sand thatτπs|F is strictly coarser than (τ1)sπ.

4. Sequential Spaces of Continuous Functions

Let X be a zero-dimensional Hausdorff topological space and let β0X be its Banaschewski compactification (see [1]). As in [1], v0X is the set of all x∈β0X with the following property: For every sequence (Vn) of neighborhoods ofxinβ0X we have∩n=1Vn∩X 6=∅. By [1, Theorem 9],v0X is the N-repletionvNX of X. Let E be a Haudorff locally convex space over K and letC(X, E) be the space of all continuous E-valued functions on X. We will denote by Cs(X, E) (resp.

Cc(X, E)) the space C(X, E) equipped with the topology of simple convergence (resp. of compact convergence). Iff is a function fromX toE,pa seminorm on E andAa subset ofX, we define

kfkA,p= sup{p(f(x)) :x∈A}.

Let now p be a non-zero non-Archimedean continuous seminorm on E and set Gp={p(s) :s∈E}. OnGp we consider the ultrametricddefined by

d(a, b) =

0, ifa=b max{a, b}, ifa6=b.

Under this metric,Gpbecomes a real-compact, strongly ultraregular, non-compact topological space and so uGpX = v0X = vNX (see [1, Theorem 9]). If f is in C(X, E), then the function

fp:X →Gp, fp(x) =p(f(x)),

is continuous and so it has a continuous extension ˜fp to all ofv0X.

Proposition 4.1. If Cs(X, E)or Cc(X, E)is sequential, then E is sequential andX isN-replete.

Proof. Let W be a convex sequential neighborhood of zero in E. Let F = Cs(X, E) (resp. F = Cc(X, E)) and suppose thatF is sequential. Let x0 ∈ X. The set

V ={f ∈F :f(x0)∈V}

is a convex sequential neighborhood of zero in F. Since F is sequential, there exists a finite (resp. compact) subsetD ofX and a continuous seminormq onE such that

(*) {f ∈F :kfkD,q≤1} ⊂V .

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It follows from (*) that

{s∈E:q(s)≤1} ⊂V and soV is a neighborhood of zero inE.

To prove thatX isN-replete, suppose that there existsy0∈v0X\X. Letpbe a continuous non-zero seminorm onE. We defineqonF byq(f) = ˜fp(y0). It is easy to see thatq is a non-Archimedean seminorm on F. Moreover, q is sequentially continuous.

In fact, let fn →0 in F and suppose thatq(fn)6→0. Going to a subsequence if necessary, we may assume thatq(fn)> ε >0 for alln. Set

Vn={y∈u0X: ( ˜fn)p(y)> ε}.

EachVn is a neighborhood ofy0inv0X and so ∩n=1Vn∩X 6=∅. Letz∈Vn∩X for all n. For this z, we have p(fn(z)) > ε for all n, which contradicts the fact thatfn→0 inF.

Proposition 4.2. IfE is metrizable, then the following assertions are equiva- lent:

(1)Cs(X, E)is bornological.

(2)Cs(X, E)is sequential.

(3)Cc(X, E)is bornological.

(4)Cc(X, E)is sequential.

(5)Cs(X,K)is bornological.

(6)Cs(X,K)is sequential.

(7)Cc(X,K)is bornological.

(8)Cc(X,K)is sequential.

(9)X isN-replete.

Proof. Since every bornological space is sequential, it follows from the preceding Proposition that each of the (1)–(8) implies (9). Also, by [6, Theorem 2.9], (1)⇔ (3)⇔(9) and (5)⇔(7)⇔(9). Thus the result follows.

References

1.Bachman G., Beckenstein E., Narici L. and Warner S.,Rings of continuous functions with values in a topological field, Trans. Amer. Math. Soc.204(1975), 91–112.

2.Govaerts W.,Bornological spaces of non-Archimedean valued functions with the compact open topology, Proc. Amer. Math. Soc.78(1980), 132–134.

3.Caenepeel S. and Schikhof W. H.,Two elementary proofs of Katsaras’ theorem on p-adic compactoids, Proc. Conf. onp-adic Analysis, Hengelhoef, 1986, 41–44.

4.Katsaras A. K.,On the topology of simple convergence in non-Archimedean function spaces, J. Math. Anal. Appl.111, No 2 (1985), 332–348.

5. , Spaces of non-Archimedean valued functions, Bolletino U.M.I. (6) 5-B (1986), 603–621.

6. ,Bornological spaces of non-Archimedean valued functions, Proc. Kon. Ned. Akad.

Wet.A 90, No 1 (1987), 41–50.

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