PONTRYAGIN DUALITY FOR TOPOLOGICAL ABELIAN

GROUPS

SALVADOR HERNANDEZ

Abstract. A topological abelian groupG is Pontryagin reflexive, or P-reflexive

for short, if the natural homomorphism of G to its bidual group is a topologicalisomorphism. In this paper we look at the question, set by Kaplan in 1948,

of characterizing the topological Abelian groups that are P-reflexive. Thus,

we find some conditions on an arbitrary group G that are equivalent to theP-reflexivity of G and give an example that corrects a wrong statement ap-

pearing in previously existent characterizations of P-reflexive groups. Some

other related properties of topological Abelian groups are also considered.

1. introduction

In this paper we look at the question, set by Kaplan in 1948, of characterizingthe topological Abelian groups for which Pontryagin duality holds. Some otherrelated properties of topological groups are also considered. To begin with we shalldescribe briefly the elements of the theory.

Let (G, �) be an arbitrary topological abelian group. A character on (G, �) is acontinuous function � from G to the complex numbers of modulus one T such that�(xy) = �(x)�(y) for all x and y in G. The pointwise product of two characters

is again a character, and the set G of all characters is a group with pointwise

multiplication as the composition law. If G is equipped with the compact open

topology, it becomes a topological group (G, �) which is called the dual group of(G, �). There is a natural evaluation homomorphism (not necessarily continuous)

eG : G −→ ˆG of G to its bidual group. A topological abelian group (G, �) satisfies

Pontryagin duality or, equivalently, is Pontryagin reflexive (P-reflexive for short,)if the evaluation map eG is a topological isomorphism onto. If eG is just bijective,we say that G is P-semireflexive.

The Pontryagin-van Kampen theorems says that every locally compact abeliangroup satisfies Pontryagin duality. The notion of duality, in one form or another,has always been a crucial point in mathematics and Pontryagin duality is one bestexample of that; (for the significance of Pontryagin duality for the rest of mathemat-ics, one should consult Mackey [12].) This important result has been subsequentlyextended to more general classes of topological groups. Kaplan [8, 9] proved thatthe class of P-reflexive groups is closed under arbitrary products and direct sumsand he set the problem of characterizing the class of topological abelian groups forwhich the Pontryagin duality holds. Smith [15] proved that the additive groups

1991 Mathematics Subject Classification. Primary 05C38, 15A15; Secondary 05A15, 15A18.

Key words and phrases. Keyword one, keyword two, keyword three.Research partially supported by Spanish DGES, grant number PB96-1075, and Fundacio Caixa

Castello, grant number P1B98-24.

1

2 SALVADOR HERNANDEZ

of Banach spaces and reflexive locally convex vector spaces are P-reflexive. Morerecently, Banaszczyk [2] has considered this question within the class of nucleargroups. Pontryagin duality for free topological abelian groups and spaces of con-tinuous functions has been investigated in [4, 6]. Motivated by Kaplan’s question,Venkatamaran [19] found a characterization of the groups satisfying Pontryaginduality, however, this characterization is very technical and, moreover, contains awrong statement. A nicer characterization of P-reflexive groups was found by Kye[11] for the class of additive groups of locally convex spaces but again the character-ization is incomplete since it contains a wrong statement which is analogous to theone in Venkatamaran’s result. In both cases it is the P-semireflexivity of the groupsconcerned what is really characterized. But, even in this case, a simpler charac-terization is desirable for general topological abelian groups. Recent contributionsrelated to the characterization of P -reflexive groups are [1, 4, 6, 16].

In the present paper we show a characterization of the topological abelian groupsthat satisfy Pontryagin duality and we give an example that corrects the wrongstatement that appears in the characterizations given by Venkatamaran [19] andKye[11]. Our main tool has been the notion of ”groups in duality”. This is a crucialconcept of Functional Analysis that was first introduced by Varopoulos (cf. [17])in the context of topological abelian groups.

2. groups in duality

Definition 1. (Varopoulos) Let G and G′ be two abelian groups then we say thatthey are in duality if and only if there is a function

⟨., .⟩ : G×G′ −→ T such that

⟨g1g2, g′⟩ = ⟨g1, g

′⟩ ⋅ ⟨g2, g′⟩

⟨g, g′1g′2⟩ = ⟨g, g′1⟩ ⋅ ⟨g, g′2⟩for all g1, g2, g ∈ G and g′, g′1g

′2 ∈ G′ and it holds: (i) if g ∕= 0G, the neutral

element of G, then there exists g′ ∈ G′ such that ⟨g, g′⟩ ∕= 1; and (ii) if g′ ∕= 0G′ ,the neutral element of G′, there exists g ∈ G such that ⟨g, g′⟩ ∕= 1.

Definition 2. (Varopoulos) If we have a duality ⟨G,G′⟩ we say that a topology �on G is compatible with the duality when (G, �)ˆ = G′.

Thus we can say that a topological abelian group (G, �) is maximally almost

periodic (MAP) when the topology � is compatible with the duality ⟨G, G⟩ which

is defined by ⟨g, �⟩ = �(g) for all g ∈ G and � ∈ G. Notice that every P-reflexivegroup must be maximally almost periodic.

For a given duality � = ⟨G,G′⟩ we may associate two canonical topologies on Gand G′ respectively. The topology w(G,G′) on G is the weak topology generatedby all the elements in G′ considered as continuous homomorphisms into T. Thetopology w(G′, G) is defined similarly. Both topologies are totally bounded andcompatible with the given duality, therefore, their completions are compact groupsdenoted by b(G,G′) and b(G′, G) respectively (cf. [17].)

For any subset A in G, we denote by A� to the set of all g′ ∈ G′ such thatRe⟨g, g′⟩ ≥ 0 for all g ∈ A, where Rez means the real part of the complex numberz. Analogously, if B′ ⊂ G′, then B′

� is the set of all g ∈ G such that Re⟨g, g′⟩ ≥ 0 forall g′ ∈ B′. These two operators behave in many aspects like the polar operator invector spaces. For instance, it is easily checked that ((A�)�)� = A� for any A ⊂ G

PONTRYAGIN DUALITY 3

and ((B′�)�)� = B′

� for any B′ ⊂ G′. Given an arbitrary subset A in G, we definethe �-convex hull of A, denoted co�(A), as the set (A�)�. A set A is said �-convexwhen it coincides with its �-convex hull. It is easy to see that different dualitiesgive place to different ”convex” hulls; for example, if � = ⟨ℚ,ℝ⟩ and � = ⟨ℝ,ℝ⟩and both are equipped with the standard bilinear mapping defined by ⟨a, b⟩ = abthen, defining L = [−1, 1] ∩ℚ, it holds that co�(L) = L but co�(L) = [−1, 1].

Given a duality � = ⟨G,G′⟩, if � is a topology on G, we say that � is locally�-convex when there is a neighborhood base of the identity consisting of �-convexsets. It is readily verified that w(G,G′) is the weakest locally �-convex topologycompatible with the duality �.

In [17] Varopoulos found the following characterization of a compatible topology.

Theorem 1. (Varopoulos) Let � = ⟨G,G′⟩ be a duality and let us denoted by � theduality ⟨G, b(G′, G)⟩. Then a necessary and sufficient condition for an arbitrarytopology � on G to be compatible with the duality � is that, for every neighborhoodU of the identity in � , it holds U� = U� and � ≥ w(G,G′).

As a consequence of this result it follows the following simple characterizationof continuity.

Corollary 1. With the same hypothesis as in the Theorem above, it holds that anecessary and sufficient condition for an arbitrary element � ∈ b(G′, G) to be in G′

is that {�}� is a neighborhood of the identity in some topology � on G compatiblewith �.

Proof. If U := {�}� is a neighborhood of the identity in some topology � on Gcompatible with �, then � ∈ U� = U� ⊂ G′.

Conversely, if � ∈ G′, then {�}� = {�}� is a neighborhood of the identity in thetopology w(G,G′). □

3. convex compactness property

A locally convex vector space E has the convex compactness (cc) property whenthe closed absolutely convex hull of every compact subset K, of E, is compact. Thisis a well known notion of the theory of locally convex vector spaces which appearsin relation to questions dealing with duality and completeness. For example, it iscrucial in the characterization of semi-reflexivity for the additive groups of locallyconvex vector spaces (see [6]) and is equivalent to completeness for metrizable locallyconvex vector spaces (see [13].) It would be desirable to extend this kind of resultsto general topological abelian groups but one basic obstruction to accomplish itis the lack of a sufficiently general extension of the Hahn-Banach theorem to thiswider context. So it is not possible in general to translate most results that holdfor locally convex vector spaces to general topological abelian groups in a simpleway as we shall see below. However, we shall show next that with an appropriatemodification of several basic tools used in the theory of locally convex vector spacesit is still possible to obtain an extension of that theory to a more general context.

Here on, given any topological abelian group G, the duality ⟨G, G⟩ is denotedby p. The group is said locally quasi-convex if it is MAP and admits a baseof neighborhood of the identity consisting of p-convex sets. We say that a locallyquasi-convex group G has the convex compactness (cc) property when the p−convexhull of every compact subset of G is compact as well (see also [3].) Our first result

4 SALVADOR HERNANDEZ

shows that the cc property characterizes completeness for metrizable locally quasi-convex groups what generalizes the analogous result given by Ostling and Wilanskyfor locally convex vector spaces (cf. [13].) In the sequel we identify T to the interval[−1/2, 1/2) in order to use the additive notation. With this notation, given a duality� = ⟨G,G′⟩ and any subset A in G, we have that A� = {g′ ∈ G′ : ∣⟨g, g′⟩∣ ≤ 1/4for all g ∈ A}

Theorem 2. Let G be a metrizable locally quasi-convex group. Then G is completeif and only if G has the cc property.

Proof. Necessity is clear.Sufficiency : Let {an}n<! be a Cauchy sequence in G. Clearly the sequence

{kan}n<! is also Cauchy for every k < !. Now, since G is metrizable and locally

quasi-convex it follows that the dual group G is hemicompact and that G is topo-

logically embedded inˆG (see Corollary 2 below.) So that we may assume that G

has the compact open topology on G. Hence, given any m < !, there is n(m) < !such that

∥ an(m) − an(m)+p ∥Km:= sup{∣an(m)(�)− an(m)+p(�)∣ : � ∈ Km} < 1/22m

where {Kn}n<! is an increasing sequence of compact subsets of G that eventuallycontains any compact subset of it. Moreover, we may also assume that

∥ k(an(m) − an(m)+p) ∥Km< 1/22m

for all integer k such that 1 ≤ k ≤ 2m+1.We now define bm := an(m) for all m < !. The sequence {k(bm+1−bm)}m<! con-

verges to 0G for all k < !. Consider now K := {b1, 2b1}∪m<!∪2m+1

k=1 {k(bm+1−bm)}.It is not difficult to check that K is a compact subset in G. Now consider the du-

ality p = ⟨G, G⟩ and take any character � in Kp. Since b1 and b2 belong toK, by [2, lemma 1.2], it follows that ∣�(2b1)∣ = ∣2�(b1)∣ ≤ 1/4 and, as a con-sequence, ∣�(b1)∣ ≤ 1/(2 ⋅ 4). Analogously, we have {bm+1 − bm, ..., k(bm+1 −bm), ..., 2m+1(bm+1−bm)} ⊂ K, so that, using [2, lemma 1.2] inductively, we obtain∣�(bm+1− bm)∣ ≤ 1/(2m+1 ⋅ 4) for all m < !. Then we have that bm+1 = b1 + (b2−b1) + ...+ (bm+1 − bm) with ∣�(b1)∣ ≤ 1/(2 ⋅ 4) and ∣�(bk+1 − bk)∣ ≤ 1/(2k+1 ⋅ 4) for1 ≤ k ≤ m. Hence, ∣�(bm+1)∣ ≤ ∣�(b1)∣+

∑mk=1 ∣�(bk+1 − bk)∣ ≤ 1

4

∑mk=1

12k+1 <

14 .

This means that bm+1 ∈ cop(K) for all m < !. Since cop(K) is compact by hy-pothesis, there must be a point b in cop(K) which is a closure point of the sequence{bm}m<!. Now, {bm}m<! is a subsequence of the Cauchy sequence {an}n<! and,as a consequence, it follows that the latter converges to b what completes theproof. □

Remark 1. It is not difficult to extend the result above to almost metrizable groupsusing the fact that every almost metrizable group G contains a compact subgroup Hsuch that the quotient group G/H is metrizable (see [14].)

It is known that the convex compactness property characterizes semireflexivityin the context of locally convex vector spaces (see [11] or [6]) but unfortunatelythis characterization does not extend to general topological abelian groups. In fact,there are complete metrizable abelian groups that are not P-semireflexive (see [1]and [3].) In order to extend and unify the duality methods that hold for LCAgroups and locally convex vector spaces to a wider context we introduced in [5] the

PONTRYAGIN DUALITY 5

notion of g-group motivated by Grothendieck’s characterization of completenessfor locally convex vector spaces. In the next section we make use of this notion tocharacterize the reflexivity of general topological abelian groups.

4. g-groups

Given a locally quasi-convex group (G, �), let ℰ be the set of all equicontinuous

subsets of G with respect to (G, �). We denote by G the collection of all those

characters on G whose restriction to each E ∈ ℰ are w(G,G)-continuous. It is clear

that G is a group containing G as a subgroup. G is equipped with the topology �whose neighborhood base of the identity consists of sets of the form Eg with E ∈ ℰ ,

where g denotes the duality ⟨G, G⟩. It is easy to see that � is a group topology.

Furthermore, it is a locally quasi-convex topology on G. The fact that every suchE is equicontinuous implies finally that �∣G = � .

We say that (G, �) is the g-extension of (G, �). When G coincides with thecompletion G of the group, it is said that (G, �) is a g-group. We have the followingalgebraic inclusions:

G ⊂ G ⊂ GG ⊂ ˆ

G ⊂ G.

So, for a complete g-group, all these groups coincide and, therefore, it is P -semireflexive.The class of g-groups contains LCA groups, additive groups of locally convex vectorspaces and nuclear groups among others.

Let (X, �) be any topological space, and let k(�) denote the k-extension of � ;that is, the largest topology on X coinciding on compact sets with � .

Remark 2. Applying some well known results (cf. [20]), it is easily verified thatif Cc((X, �),T) is the group of all continuous functions of (X, �) into the one-dimensional torus, equipped with the compact open topology, then the completion ofthis group may be identified with Cc((X, k(�)),T).

Lemma 1. Let (G, �) be any topological abelian group and let (G, �) denote the

completion of the dual group. Then (G, �) may be embedded as a closed subgroupof Cc((G, k(�)),T).

Proof. If � is an arbitrary element in G then it is the limit of a net {�n} on G

which converges in the compact open topology. Since the group (G, �) is obviously

embedded in Cc((G, �),T), it follows that � belongs to C((G, k(�),T). That G is atopological subgroup of Cc((G, k(�)),T) is clear by the preceding remark. □

Now we are in position of presenting the main results of this paper. In the sequel,

if (G, �) denotes the completion of (G, �) then q designates the duality ⟨G, G⟩.

Theorem 3. Let (G, �) be a topological abelian group. Then (G, �) is P -reflexiveif and only if the following conditions hold:

(1) (G, �) is locally quasi-convex;(2) cop(K) = cog(K) for all compact subset K in (G, �);

(3) for any compact subset L in (G, �) it holds that cop(L) is w(G,G)-compact

6 SALVADOR HERNANDEZ

(4) let B be any subset of G such that cop(B) is closed in G and, for everycompact subset K in (G, �) with K containing the identity element 0G ofG, Bp∩K is a neighborhood of 0G relative to K, then Bp is a neighborhoodof 0G in (G, �).

Proof. Necessity: If the group (G, �) is P -reflexive then it is obvious that condition1. is satisfied.

Let K be a compact subset of (G, �), then Kp is a � -neighborhood of the identity

in G. By Theorem 1 and the reflexivity of (G, �) it follows that cob(K) = (Kp)b =

(Kp)p = cop(K), with b = ⟨b(G, G), G⟩. Hence, cop(K) ⊂ cog(K) ⊂ cob(K) =cop(K) and 2. is verified.

Let L be a compact subset of (G, �), since the group (G, �) is P -reflexive, it is

equipped with the compact open topology on (G, �). Thus, every compact subsetof the dual group is � -equicontinuous on G and, therefore, Lp is a neighborhood of

0G in (G, �). By Theorem 1 we have that cop(L) = cob(L), with b = ⟨G, b(G,G)⟩,therefore, cop(L) is w(G,G)-compact and this proves 3.

Suppose that B is a subset of G such that cop(B) is closed in G and, for everycompact subset K in (G, �) with K containing the identity element 0G of G, Bp∩Kis a neighborhood of 0G relative to K. Then it is proved as in [18, th. 5.3] that

coq(B) is equicontinuous on the topological space (G, k(�)). By lemma 1, (G, �) isa closed subgroup of Cc((G, k(�)),T) which is complete. Thus, by Ascoli theorem,

it follows that coq(B) is a compact subset of (G, �). As a consequence, cop(B) is a

compact subset of (G, �) and Bp = (cop(B))p is a neighborhood of the identity in(G, �) and 4. is satisfied.

Suficciency: Firstly we prove that (G, �) is P -semireflexive. Suppose that � is a

character of (G, �). Then � ∈ G and Corollary 1 shows that {�}g is a neighborhood

of the identity in (G, �). Hence, there is a compact subset K in (G, �) such thatKp ⊂ {�}g. Then cog({�}) ⊂ cog(K) = cop(K) ⊂ G. This implies, with somenotational abuse, that � ∈ G and, therefore, the (G, �) is P -semireflexive.

Let us see now that the evaluation map e : (G, �) −→ (G, ˆ�) is a topologicalisomorphism. Firstly, it is easily verified that e is open and one-to-one as a directconsequence of being (G, �) locally quasi-convex. Thus only the continuity of e

need be checked. Suppose that K is a compact subset in (G, �) and consider Kp

which is a ˆ� -neighborhood of the identity in G. If L is a compact subset in (G, �),

then cop(L) = cog(L) which is ˆ� -compact, therefore, L is ˆ� -compact as well and

�∣L = ˆ� ∣L. This implies that Kp ∩ L is a � -neighborhood of the identity in every

compact subset L containing 0G. On the other hand, by 3., cop(K) is w(G,G)

compact and, a fortiori, is closed in b(G,G). Clearly, G is contained in b(G,G) and,

therefore, cop(K) is closed in G. Hence we may apply 4. to obtain that Kp is a� -neighborhood of the identity in G. This fact clearly is equivalent to the continuityof the evaluation map e and we are done. □

If (G, �) is a g-group the duality g = ⟨G, G⟩ coincides with the duality p = ⟨G, G⟩where G denotes the completion of the group G. Hence, as a consequence of theTheorem above, we obtain that a g-group (G, �) is P -semireflexive if and only ifcop(K) = cop(K) for all compact subset K in (G, �). Thus, it follows that every

PONTRYAGIN DUALITY 7

complete g-group is P -semireflexive. When we consider the additive group of alocally convex vector space, the Hahn-Banach theorem permits to improve thischaracterization of semireflexivity by the property that closed absolutely convexhulls of compact sets are compact as well. However, We do not know whether ornot the cc property is equivalent to P -semireflexivity for general g-groups.

As a consequence we obtain the following Corollary.

Corollary 2. If (G, �) is a complete g-group which is a topological k-space thenthe group is P -reflexive.

Since LCA groups, nuclear groups and the additive groups of locally convexvector spaces are g-groups, the Corollary above extends and unifies the equivalentduality results obtained separately for each of these classes of topological groups.

Obviously, a question that one may set when dealing with Pontryagin duality isthat of finding out an intern characterization of the topological groups that satisfyPontryagin duality. This goal has been accomplished for some specific classes oftopological groups, for instance in [4]. In general, if that intern characterizationdoes exist, it will not be an easy one with all probability since it was proved in [6]that the question of characterizing for what topological spaces X, the correspondingadditive groups of the rings of all continuous functions Cp(X), equipped with thepointwise open topology, are P -reflexive, is undecidable in ZFC.

5. some examples

We now establish the independence of the four conditions in Theorem 3.

Example 1. Let G be the additive group of Lp[a, b] for 0 < p < 1. Then G isminimally almost periodic, what means that has no non-trivial character. Thereforethe group trivially satisfies 2.,3. and 4. but not 1.

Example 2. Let G be the additive group of a non complete metrizable locally convexspace. It is easy to check that satisfies 1., 3. and 4. but not 2.

Example 3. Let � be a cardinal number such that cf(�) > !. Let X� be atopological space such that X� = D ∪ {∞}, ∣D∣ = �, all points of D are isolated,and the neighbourhoods of ∞ are of the form E ∪ {∞}, where ∣D∖E∣ < �. When� = (2!)+, the additive group of Cp(X�), the space of all real-valued continuousfunctions on X� equipped with the pointwise convergence topology, is P-reflexive(cf. [6, Ex. 3.11]). If we take G as the additive group of L(X�), the free locallyconvex vector space associated to X�, then G satisfies 1., 2. and 3. but not 4.

Proof. Indeed, using the fact that Cp(X�) is P -reflexive it is not difficult to verify1., 2. and 3. So we only check that 4. does not hold. Let fd ∈ C(X�) defined byfd = 1 and fd(x) = 0 for all x ∕= d. Then it is readily seen that K = {fd : d ∈ D}is a compact subset of Cp(X�) which is not equicontinuous at ∞ ∈ X�. The P -reflexivity of Cp(X�) implies that cop(K) is compact. On the other hand, X� is aP -space what means that compact subsets of L(X�) are supported on finite subsetsof X�. Thus Kp∪C is a neighbourhood of 0G relative to C for every compact subsetK ∈ G containing the identity of G. However Kp cannot be a neighbourhood of 0Gin G since it is not equicontinuous on X. □

Example 4. Let X be any topological space which is a P -space and such that Cp(X)is not P -semireflexive (cf. [6, Ex. 3.9]) and let G be the additive group of free vector

8 SALVADOR HERNANDEZ

space L(X) equipped with the Mackey topology � of the pair dual ⟨L(X), C(X)⟩ (cf.[10, 21.4]). Then G satisfies 1., 2. and 4. but not 3.

Proof. Indeed, It is easy to check that every �-compact subset of L(X) has finite

support on X and, as a consequence, the group (G, �) may be identified with Cp(X).Then 1., 2. and 4. derive from the fact that G is equipped with the Mackey topology

of a dual pair: if B is a subset of G = Cp(X) that holds the hypothesis stated in 4.,then cop(B) is a compact absolutely convex subset of Cp(X) and, therefore, Bp is aneighbourhood of the identity in G = (L(X), �). On the other hand, since Cp(X)is not P -semireflexive contain compact subsets K such that cop(K) is not compact.This proves that 3. does not hold. □

Kye has characterized the continuity of the evaluation monomorphism eE :−→ ˆE

for the additive group of an arbitrary locally convex vector space by the conditionthat each closed convex balanced set which is a neighbourhood of zero in the k-topology is a neighbourhood of zero in the original topology. And this is basicallyVenkatamaran’s characterization for general topological Abelian groups althoughwith some necessary changes in the terminology. We finish this paper by giving anexample of a topological group that is P -reflexive and does not hold either of thecharacterizations of reflexivity stated by Venkatamaran (cf. [19]) and Kye (cf. [11])respectively.

Example 5. Let X be a non-discrete topological space such that the group Cp(X) isP -reflexive (cf. [6, Ex. 3.11]) and let (L(X), �p) be its dual group equipped with thecompact open topology on Cp(X). Then there exists a subset F in Cp(X) satisfyingthat F p ∩K is a neighborhood of the identity for every compact in (L(X), �p) con-taining it but such that F p is not itself a neighborhood of the identity in (L(X), �p)(see [19, 11].)

Proof. Consider the dualities p = ⟨Cp(X), L(X)⟩ and q = ⟨ℝX , L(X)⟩ and takeL = [−1, 1]X which is a compact subset of ℝX . Then Lq is a neighborhood of theidentity in (L(X), �q) and, since L is compact and absolutely convex, it follows thatcoq(L) = L. On the other hand, it is easily verified that L ∩ Cp(X) is dense in L.Hence, [L∩Cp(X)]p = [L∩Cp(X)]q = Lq. If we define F = L∩Cp(X), we has justverified that F p = Lq is a p-convex (resp. character closed in [19]) subset of L(X).Now, let K be a compact subset of (L(X), �p). By [6, Prop. 3.1], it follows thatsup p K is finite; that is, there is n < ! such that K ⊂ ℝn. Obviously, Lq = F p is aneighborhood of zero in the l1-norm of ℝn and, therefore, F p∩K is a neighborhoodof the identity for every compact subset K in (L(X), �p).

Suppose that F p = is a neighborhood of the identity in (L(X), �p) then cop(F )

must be ˆ�p-compact subset of Cp(X). By the reflexivity of this group, we have thatcop(F ) must be compact in the standard topology of Cp(X). But this is impossiblebecause cop(F ) = coq(L) ∩ C(X) = L ∩ C(X) and it is clear that the latter, beingdense in L, may not be compact. □

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PONTRYAGIN DUALITY 9

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Universidad Jaume I, Departamento de Matematicas, Campus de Riu Sec, 12071 Castellon(Spain)

E-mail address: hernande@mat.uji.es