REPRESENTATION THEOREMS OF LINEAR OPERATORS ON P-ADIC FUNCTION SPACES

Let X be a 0-dimensional Hausforff topological space, E, F nonarchimedean Banach spaces and Cb(X,E) the space of all continuous E-valued functions on X provided with two strict topologies. In this paper we show that every F−valued linear operator which is strictly continuous can be represented by a certain L(E,F )−valued measure defined on the ring of all clopen subsets of X. ∗This work is partially supported by Fondecyt No. 1020288 98 José Aguayo, Elsa Chandía and Jacqueline Ojeda


Introduction and notations
By the classical Riesz Representation Theorem, a linear functional u on the space of continuous real functions on a compact Hausdorff space X, is continuous for the topology of uniform convergence if, and only if, there exists a bounded regular Borel measure m on X such that u (f ) = R fdm.The Riesz Representation Theorem has been extended to many other spaces (see [8]) and linear operators instead of linear functional (see [4]).The relation between vector measures, linear operators and strict topologies in the classical case have been studied by several authors (see [1] [4]).Analogous situation in the non-archimedean case is studied in [5].
This paper is devoted to extend the work given in [5] for another two strict topologies.Throughout this work, X will be a zero dimensional Hausdorff topological space, K a complete non-archimedean valued field with nontrivial valuation and E, F non-archimedean Banach spaces.
We will denote by C b (X, E) the space of all E−valued bounded and continuous functions on X and by C rc (X, E) the subspace of C b (X, E) of those functions whose image of X are relatively compact.If E = K, we will write C b (X) and C rc (X) respectively.
We will denote by β • X the Banaschewski compactification of X [7] and understand by b f the unique continuous extension of f to β • X.For A ⊂ X, we will denote by For an E−valued function f on X and A ⊂ X, we will denote Let S(X) be the collection of all clopen subsets of X.An L(E, F )− valued set function m on S(X) is said to be a measure if: We will denote by M (X, L(E, F )) the space of all these measures.For m ∈ M (X, L(E, F )) and A ∈ S(X), we define In order to introduce strict topologies, we will denote by Ω the collection of all compact subsets of β • X \ X and by The strict topology β (β u ) on C b (X, E) is the inductive limit of the locally convex topologies β Q , where β Q is generated by the family of seminorms [5] , [2,3]).
Next, we will define the integrability of an E−valued function f on X with respect to a m ∈ M (X, L(E, F )).For A ∈ S(X), is a clopen partition of A and x i ∈ A i .We will introduce the following relation: α 1 ≥ α 2 iff the partition of A in α 1 is a refinement of the partition of A in α 2 .We will denote by Ω A the collection of all these α.Ω A will become to be a directed set.For f, m and α Note that (f, m) ∈ F.
We will say that f is m−integrable over A if lim α (f, m) exists; in such a case, we will denote this limit by If A = ∅, then we will define R ∅ fdm = 0.For A = X, we will simply write R fdm.It is easy to see that if f is m−integrable over X, then f is m−integrable over every A ∈ S(X).
We will present the following very well-known technical result.
Lemma 1.Let ε > 0 and f ∈ C rc (X, E).Then, there exist disjoint clopen sets A 1 , A 2 , ..., A n covering X and elements e 1 , e 2 , ..., e n of E such that °°°°°f where X A i e i (x) = e i , if x ∈ A i and the null element θ of E otherwise.
In this section we will study the relation between measure theory and F −valued continuous linear operators on C rc (X, E).We will denote by L(C rc (X, E), F ) the space of all these operators. Proof.
Without loss of generality, we can assume that A = X and kmk (X) ≤ 1.Let µ ∈ K, with 0 < |µ| ≤ 1, and ε > 0. We take ν ∈ K such that 0 Therefore, 1.If the valuation of K is dense or the valuation is discrete and kEk ⊂ |K| , then Proof.
1.) We can assume that 0 < kf k A < ∞, since otherwise, the statement is trivial.Under the denseness conditions, for each ε > 0, there exists λ ∈ K such that

2.) If we consider the norm kek
Remark 4 : The previous lemma proves that if m ∈ M (X, L(E, F )), then the linear operator fdm is a T u −continuous linear operator, where T u denotes the uniform convergence topology on C rc (X, E).
Proof.For each A ∈ S(X), we define Since T is bounded, with bound M > 0, we have ).We claim that the set-function is a measure.In fact, trivially m is well-define and finitely additive.To prove that {m(A) : A ∈ S(X)} is equicontinuous, take ε > 0 and choose On the other hand, by the facts that h{X A e : e ∈ E, A ∈ S(X)}i is T u −dense in C rc (X, E) (see Lemma 1) and both T m and T are T u −continuous, we get Corollary 6 : The mapping is an algebraic isomorphism.

τ and u-additive measures.
This section will devote to study certain class of members of M (X, L(E, F )) and study the behavior of the associated F −valued continuous linear operators given in the previous section.
2. m is u-additive if for each clopen partition {U α } α∈I of X, we have where the limit has to be taken over the directed set of all finite subsets J ⊂ I.
Proof.Let m ∈ M τ (X, L(E, F )) and (U α ) α∈I be a clopen partition of X.
For any finite subset J of I, we define the decreasing net {A J } J , where In the previous section we proved that if f ∈ C rc (X, E) and m ∈ M (X, L(E, F )), then f is m−integrable over any A ∈ S(X).The next theorem will extend this result.

Proof.
Without loss of generality, we can assume that kf k ≤ 1, and kmk (X) ≤ 1.For a given ε > 0, we define the following equivalence relation Each B J is clopen and B J ↓ ∅.Since m is u− additive, we have that there exists a finite subset J • of I such that if J is another finite subset of I with J • ⊆ J, then kmk (B J ) ≤ ε.For such a J, we define the following functions Let us consider the finite clopen partition {A j : j ∈ J} ∪ {B J } of X and take {D 1 , , ...D n } a refinement of {A j : j ∈ J}∪{B J } j .If we choose y i ∈ D i , then g J (y i ) = f (x j ) and h J (y i ) = 0, if D i ⊆ A j for some j ∈ J, or g J (y i ) = 0 and h J (y i ) = f (x j ), for some j / ∈ J, if On the other hand, Remark 10 : From Prop. 8, if m is τ −additive measure and f ∈ C b (X, E), then f is m-integrable over A. In [6] , it has been proved that M t (X, L(E, F )) and the space of all F −valued and β • -continuous linear operators on C b (X, E) are algebraically isomorphic.The next theorems will show similar results for M τ (X, L(E, F )) and M u (X, L(E, F )).
Theorem 11 : If T ∈ L(C b (X, E), F ), then the following statements are equivalent: 1. T is β-continuous.
2. The associated measure m is τ −additive.
Proof.1.) ⇒ 2.) Let {A α } α∈I be a net of clopen subsets of X such that A α ↓ ∅.By the continuity of T , is a β-neighborhood of 0, and by the definition of β, W is a β K -neighborhood of 0, for all K ∈ Ω.Now, since For a given ε > 0, we choose µ ∈ K, with 0 < |µ| < ε and define Therefore, since e and A are arbitrary, we have kmk (A) ≤ ε.
2.) ⇒ 1.) We will prove that Theorem 12 : If m ∈ M τ (X, L(E, F )), then the linear operator Proof.The same arguments used in the previous theorem proves this statement and then we omit the proof.
Theorem 13 :M τ (X, L(E, F )) and the space of all F −valued and βcontinuous linear operators on C b (X, E) are algebraically isomorphic.
Proof.In order to prove this theorem, we need to prove that the linear map is an algebraic isomorphism, where L β (C b (X, E), F )) denotes the space of all β-continuous linear operators.
It is easy to see that the map Ψ is linear and one to one.To prove that Ψ is onto, take and then °°T (µ −1 X A e) °°≤ 1 or equivalently km(A)ek ≤ |µ| ≤ ε.
T : C b (X, E) → F and prove that T = T m for the associated measures m.By Th. 11, m ∈ M τ (X, L(E, F )) and T = T m on C b (X, E) follows from the β−denseness of C rc (X, E) in C b (X, E) and the β−continuity of both T and T m .Theorem 14 : If m ∈ M u (X, L (E, F )) , then T m is β u -continuous.Proof.Let W = {f ∈ C b (X, E) : kT m (f )k ≤ 1} and Q ∈ Ω u .There exists a clopen partition {A i } i∈I of X such that Q ∩ A i β•X = ∅ , ∀i ∈ I.For any finite subset J of I, we defineB J = X \ S i∈J A i .Since m ∈ M u (X, L (E, F )), there exists a finite subset J • of I such that for a given δ > 0, we havekmk (B J ) ≤ δ −1 , for any finite subset J of I with J • ⊂ J.If B = i ∈ J • ∪A i , then B is clopen in X and B β • X ∩ Q = ∅.We claim that U = {f ∈ C b (X, E) : kf k ≤ δ, kf k B ≤ 1} ⊂ W. In fact, if f ∈ C b (X, E), then °°°°Z B f dm °°°°≤ kf k B kmk (B) ≤ kf k B kmk (X) ≤ 1.On the other hand,°°°°°Z X\B f dm °°°°°≤ kf k kmk (B J ) ≤ δδ −1 Q ∈ Ω u isarbitrary, the continuity of T m follows.Theorem 15 : M u (X, L(E, F )) and the space of all F −valued and β ucontinuous linear operators on C b (X, E) are algebraically isomorphic.