On the topology of compactoid convergence in non-archimedean spaces

Some of the properties, of the topology of uniform convergence on the compactoid subsets of a non-Archimedean locally convex space E, are studied. In case E is metrizable, the compactoid convergence topology coincides with the finest locally convex topology which agrees with a~E’, E) on equicontinuous sets.


Introduction
In [7J some of the properties of the topology of uniform convergence on the compactoid subsets, of a non-Archimedean locally convex space, are investigated. In the same paper, the authors defined the ~-product E~F of two non-Archimedean locally convex spaces E and F. EeF is the space of all continuous linear operators of E~ to F equipped with the topology of uniform convergence on the equicontinuous subsets of E', where E~ is the dual space E' of E endowed with the topology of uniform convergence on the compactoid subsets of E. In this paper, we continue with the investigation of the compactoid convergence topology T co. Among other things, we show that, for metrizable E, rco coincides with the topology 03C3, where 03C3 is the finest locally convex topology on E' which agrees with Q(E', E) on equicontinuous 1Key words and phrases: compactoid set, e-product, polar space, nuclear operator.
A.M.S. Subject Cfassi6cation: 46S10 sets. We also prove that 1~0 has a base at zero all sets ~°t~'~, , where W is a 03C403C3-neighborhood of zero and W03C3(E',E) denotes the 03C3(E', E)-closure of W.

Preliminaries
Throughout this paper, K will stand for a complete non-Archimedean valued field, whose valuation is non-trivial, and N for the set of natural numbers. By a seminorm, on a vector space E over K, we will mean a non-Archimedean seminorm.
Let now E be a locally convex space over K. The collection of all continuous seminorms on E will be denoted by cs(E). The algebraic dual, the topological dual, and the completion of E will be denoted by E*, E' and E respectively. A seminorm p on E is called polar if p = sup{|f| :f E E*, |f| ~ p}, where is defined by ~ f , ( x) = .
The space E is called polar if its topology is generated by a collection of polar seminorms. The edged hull Ae, of an absolutely convex subset A of E, is defined by: Ae = A if the valuation of K is discrete and Ae = a ~ > 1 ~ if the valuation is dense (see [10] 3) nuclear if there exist a null sequence in K, a bounded sequence (yn) in F and an equicontinuous sequence ( fn) in E' such that 00 Tx = 03A3 03BBnfn(x)yn f=i for all x E.
We will denote by the dual space E' of E equipped with the topology of uniform convergence on the compactoid subsets of E. The ~-product EeF, of two locally convex spaces E, F is the space of all continuous linear maps from to F endowed with the topology of uniform convergence on the equicontinuous subsets of ~E'. For other notions, concerning non-Archimedean locally convex spaces and for related results, we will refer to [10].
We will need the following Lemma 2.1 ( j7, Lemma 2. 6J ). Let E, F be Hausdorff polar quasi-complete spaces and let T : E' t-t F be a linear map. If T is continuous with respect to the weak topologies a(E', E) and Q(F, F'), then T E EeF iff T maps equicontinuofJ8 subsets of E' into compactoid subsets of F.

The topology Tq
Let E be a Hausdorff polar space. We will denote by Ta the finest locally convex topology on E' which agrees with Q(E', E) on equicontinuous sets.
It is easy to see that To is the locally convex topology which has as a base at zero all absolutely convex subsets W of E' with the folowing property: For every equicontinous subset H of E' there exists a finite subset S of E such that S° n H C W, where S° is the polar of S in E'. In case E is a normed space, 7a coincides with the bounded weak star topology bw' (see [12] or [13] ).
Since a linear functional f on E' is 03C3-continuous iff its restriction to every equicontinuous subset of E' is E)-continuous we have the following Proposition 3.1 If E is a Hausdorff polar space, then (E', 03C3)' = E. Proof. See the proof of Theorem 2 in ~5j. The  Let now q be a Ta-continuous seminorm on E' and set Wm = ~x' E E' : q~x') Each V0n is a 03C3(E', E)-compactoid and hence a TQ-compactoid since V0n is absolutely convex and 03C3 = 03C3(E', E) on V,°. Thus, for each mEN, there exists a finite Snm of E' such that Yn C co(Snm) + Wm. Now, the set S = ~m,n Snm is countable and the space [6'j is q-dense in E'. This completes the proof.
Let now E be a Hausdorff polar space and let j E E '-~ E" the canonical map. In the following Theorem, we will consider E as a vector subspace of E" identifying E with its image under the canonical map. For a subset A of E" we will denote by A° and A°°, respectively, the polar and the bipolar of A with respect to the pair E", E' >. If we consider on E" the topology of uniform convergence on the equicontinuous subsets of E', then E will be a topological subspace of E". In this case E" will have as a base at zero all sets V~ where V is a convex neighborhood of zero in E.
The proof of the next Proposition is an adaptation of the corresponding proof for normed spaces given by Schikhof , -continuous on V00, it follows that the restriction of x" to V00 is 03C3(E', E)continuous. This clearly proves that x" is continuous.
On the other hand, let x" e F n .E" and let V be a convex neighborhood of zero in E. Let |03BB| > 1 and set There exists a finite subset S of E such that The set A = co(S) is a complete metrizable compactoid in (E", v(E", E')). .
Since V00 is absolutely convex and E')-closed, it follows that (A + V00)e is Q(E", E')-closed by (11, Theorem 1.4]. Since SO n V° = (A + V)° , we get that 03BBD0 C (A + V)00 = A + V00) 00 = = A + V00)e and so D° C A + V°° C E + V°°. Since x" E D°, it follows that x" E E, which completes the proof. As we will see in the next section, if E is metrizable, then is coarser than the strong topology on E' and so in this case = E, a result proved by Schikhof in [12] for normed spaces. 4 The Topology of Compactoid Convergence For a locally convex space (E, T), we will denote by T~ the topology of compactoid convergence, i.e the topology on E' of uniform convergence on the compactoid subsets of E. We will denote by E~. By (?, 3.3~, , every equicontinuous subset of E' is Tco-compactoid. Example If E = co with the usual norm topology, then E' = loo and Teo is the topology generated by the seminorms p~, z = E ce where pz(x) = max03BA |z03BAx03BA| for x = (xn) E l~. This follows from the fact that a subset A of c0 is compactoid iff A ~ = {x ~ c0 : |xn| ~ |zn| J ~n} for some z E CO. ' Notation For a locally convex topology 03B3 on E', we will denote by 03B303C3 the locally convex topology on E' which has as a base at zero all sets of the Proof. Since Tco 03C3, we have that On the other hand, let W be a convex Tu-neighborhood of zero. If V is a polar neighborhood of zero in E and > 1, then there exists a finite subset S of E such that S0 ~ V0 C 03BB-1W. Since S0 ~ V0 = (co(S) + V)o, , it follows that 03BBW0 ~ (co(S) + = (co(S) + V)e C a (co(S) + V) (by [10, Corollary 5.8]). Thus W'° C co(S) + Y, which shows that W0 is a compactoid subset of E. Thus W°° is a CO- and so is a co-neighborhood of zero. This completes the proof. The following is a Banach-Dieudonne type Theorem for non-Archimedean spaces (see [3,Theorem 10.1]). Theorem 4.5 If (E, T) is metrizable polar space, then co = 03C3.
Proof. Let (Vn) be a decreasing sequence of convex neighborhoods of zero in E which is a base at zero and let D be a convex 03C3-neighborhood of zero in E'. Since To is the finest locally convex topology on E' which agrees with Q(E', E) on the sets V,°, n EN, we may assume that there exists (by (4, Theorem 5.2]) a sequence (Sn)~n=0 of finite subsets of E such that for Wn we have 00 D = Wp n n Wn + V0n)).
Since each Wn + V0n is 03C3(E', E)-closed and since 4Yc is also a(E', E)-closed, it follows that D = D03C3(E',E). . Now since co = it follows that 03C3 ~ co . This clearly completes the proof. 2) Is it always true that 03C3 = co ?
The following Theorem gives a necessary and sufficient condition for the topology Too to be compatible with the pair E~, E >. (2) Every closed (or equivalently weakly closed) compactoid subset of E is complete.
(3) Every closed (or equivalently weakly closed) absolutely convex subset of E is weakly complete.
Proof. First of all we observe that a compactoid subset of E is closed iff it is weakly closed and that an absolutely convex compactoid is complete iff it is weakly complete (by [10, Theorem 5. I3~ ) .
( 1 ) ~ (2) . Let A be a closed compactoid subset of E. Since co is compatible with the pair E', E >, it is the topology of uniform convergence on some special covering (by [12, Proposition 7.4}). Thus, there exists a weakly bounded, weakly complete edged subset B of E such that B° C A°. Thus , Since A°° is an absolutely convex weakly complete subset of E, it is complete and hence A is complete.
(2) ~ (3).It is trivial.  (2) If we consider on G the topology of uniform convergence on the equicontinuous subsets of E', then E is a dense topological subspace of G.
Proof. (1) Since the topology of E~ is coarser than the strong topology on E', G is a vector subspace of E". For a subset B of G we denote by B° and respectively, the polar and the bipolar of B with respect to the pair G, E' >. Let now x" E G. There exists an absolutely convex compactoid subset A of E such that A°C{x'EE: : ~x',x">~~~. If (a~ > 1, then On the other hand, if x" E 7 for some absolutely convex compactoid subset A of E, then x" E A°° and so ~ x', x" > ~ 1 for x' E A4, which implies that x" E G. (2) Since the topology of is finer then the topology ~(E', E) and since E is Haudorff and polar, it follows that E is a topological subspace of G. It only remains to show that E is dense in G. So let x" E G. By (1), x" E for some absolutely convex compactoid subset A of E. Given a convex neighborhood V of zero in E and ~a~ > 1, there exists a finite subset S of E such that A C co(S) + C co(S) + 03BB-1V00. Now x" ~ A00 ~ (co(S) + 0 3 B B -1 V 0 0 ) 00 = (co(S) + and so x" ~ 03BBco(S) + V00. (2) Assume that T is compactoid and let p E cs(E) be such that the set A = T( Vp) is compactoid in F where Y~ = {x E E : p(x) 1 ~. .
We will finish the proof by showing that is a compactoid subset of So, let B be a compactoid subset of E. Since E is polar, it has the approximation property (by [9, Theorem 5.4~), Thus there are gl, ~ . , 9" in E' and el,... , en in E such that  T''y', x > = y',T x > = y', 03A303BBnfn(x)yn > = 03A3 03BBnfn(x)y'(yn). On the f-product As it is shown in [7], the e-product of two polar complete spaces is complete. The following proposition shows that the same is true for quasicomplete spaces.  'ao(f)a9 >= f~'~(9) > ~ . It follows from this that u0(T0) ~ S°. For a Hausdorff polar space F, we denote by F the dual space of F'ẽ quipped with the topology of uniform convergence on the equicontinuous subsets of F'. It is easy to see that if u E FeE, then the adjoint u' belongs to E~. We will consider F as a topological subspace of F. Proposition 5.3 Let E, F be Hausdorff polar spaces. Then, the map u ~+ u', from F~E to E~, is linear, continuous and one-to-one.
Proof. For a convex neighborhood V of F, we will let V°° denote the bipolar of V with respect to the dual pair F, F >. . Sets of the form Y°° form a base at zero in F. Let now W and V be convex neighborhoods of zero in E and F respectively and let D = {v E EeF : If u E FeE is such that u(VO) C W, then u' E D. This proves that the map u H u' is continuous. The rest of the proof is trivial. (2) If every dosed compactoid subset of F' is complete, then D is an equicontinuoas subset of E). . (2) If every closed compactoid subset of F is complete, then F = F (by Theorem 4.7) and so the set .D' _ {u' : u E D~ is a compactoid subset of EeF by the preceding Proposition. Given a polar neighborhood W in E, the set W° is an equicontinuous subset of E' and so A = D'(WO) is a compactoid subset of F by the first part of the proof. Moreover, for u E D, we have u(Ao) C W°° = W which completes the proof of (2).
(3) The set D is a compactoid subset of FEE. Let (ua) be a Cauchy net in D. For each x' E F', the set D(x') is compactoid in E and (u03B1(x')) is a Cauchy net. By our hypothesis, the limit lim u03B1(x') exists in E. Define u : : E, u(x') = lim u03B1(x').
Claim : u E FEE. Indeed, u is linear. Also, given a polar neighborhood W of zero in E, the set B = D'(W0) is compactoid in F and D(.$°) C W. If x' E B°, there exists ao such that u(x') -E W, for a ~ao, and so u(x') E + W C W, which proves that u E FeE. If H is an equicontinuous subset of F', then there exists !30 such that C W for 03B1 03B2 03B20, and so (ua -u)(H) C W for 03B1 03B20. This proves that u03B1 ~ u in FeE and the result follows. for n > mo. This clearly completes the proof.