P-adic Spaces of Continuous Functions II

Necessary and sufficient conditions are given so that the space C(X, E) of all continuous functions from a zero-dimensional topological space X to a nonArchimedean locally convex space E, equipped with the topology of uniform convergence on the compact subsets of X, to be polarly absolutely quasi-barrelled, polarly אo-barrelled, polarly `∞-barrelled or polarly co-barrelled. Also, tensor products of spaces of continuous functions as well as tensor products of certain E′-valued measures are investigated.


Introduction
This paper is a continuation of [3].Let K be a complete non-Archimedean valued field and let C(X, E) be the space of all continuous functions from a zero-dimensional Hausdorff topological space X to a non-Archimedean Hausdorff locally convex space E. We will denote by C b (X, E) (resp.by C rc (X, E)) the space of all f ∈ C(X, E) for which f (X) is a bounded (resp.relatively compact) subset of E. The dual space of C rc (X, E), under the topology t u of uniform convergence, is a space M (X, E ) of finitely-additive E -valued measures on the algebra K(X) of all clopen , i.e. both closed and open, subsets of X.Some subspaces of M (X, E ) turn out to be the duals of C(X, E) or of C b (X, E) under certain locally convex topologies.In section 1, we give necessary and sufficient conditions for the space C(X, E), equipped with the topology of uniform convergence on the compact subsets of X, to be polarly absolutely quasi-barrelled, polarly ℵ obarrelled, polarly ∞ -barrelled or polarly c o -barrelled.In section 2 , we study tensor products of spaces of continuous functions as well as tensor products of certain E -valued measures.We refer to paper [3] for the notations used in the paper as well as some preliminaries needed for the paper.

Barrelledness in Spaces of Continuous Functions
We will denote by C c (X, E) the space C(X, E) equipped with the topology of uniform convergence on compact subsets of X.By M c (X, E ) we will denote the space of all m ∈ M (X, E ) with compact support.The dual space of C c (X, E) coincides with M c (X, E ).Recall that a zero-dimensional Hausdorff topological space X is called a µ o -space (see [1]) if every bounding subset of X is relatively compact.We denote by µ o X the smallest of all µ o -subspaces of β o X which contain X.Then X ⊂ µ o X ⊂ θ o X and, for each bounding subset A of X, the set A βoX is contained in µ o X (see [1]).Moreover, if Y is another Hausdorff zero-dimensional space and f : X → Y , then f βo (µ o X) ⊂ µ o Y and so there exists a continuous extension f µo : Let us say that a family F of subsets of a a set Z is finite on a subset F of Z if the family of all members of F which meet F is finite.Definition 1.1.A subset D, of a topological space Z, is said to be wbounded if every family F of open subsets of Z, which is finite on each compact subset of Z, is also finite on D. If this happens for families of clopen sets, then D is said to be w o -bounded.We say that Z is a wspace (resp.a w o -space ) if every w-bounded (resp.w o -bounded) subset is relatively compact.Definition 1.2.A subset W , of a locally convex space E, is said to be absolutely bornivorous if it absorbs every subset S of E for which sup x∈S |u(x)| < ∞ for all u ∈ W o .The space E is said to be polarly absolutely quasi-barrelled if every polar absolutely bornivorous subset of E is a neighborhood of zero.Lemma 1.3.Every absolutely bornivorous subset W , of a locally convex space E, absorbs bounded subsets of E.
Proof: Let B be a bounded subset of E and suppose that W does not absorb Since B is bounded, we have that y n = λ −n x n → 0, and so u(y n ) → 0, a contradiction.

Definition 1.4.
A subset A, of a topological space Z, is called aw obounded if it is w o -bounded in its subspace topology.The space Z is said to be an aw o -space if every aw o -bounded set is relatively compact.
Proof: Assume the contrary.Then, there exists a sequence For each m ∈ H, the sequence (W n ) is finite on the supp(m) and thus m(W n ) = 0 finally, which implies that sup n | < m, h n > | < ∞ for all m ∈ H. Therefore, there exists α = 0 such that F ⊂ αD.But then for all n, which is impossible.This contradiction completes the proof.Theorem 1.6.Assume that E = {0}.If the space G = C c (X, E) is polarly absolutely quasi-barrelled, then E is polarly absolutely quasi-barrelled and X an aw o -space.
Proof: Let W be a polar absolutely bornivorous subset of E and let W o be its polar in E .Let x ∈ X and, for u ∈ E , let u x ∈ G , u x (f ) = u(f (x)).Consider the set H = {u x : u ∈ W o }, and let D = H o be its polar in G. Then D is absolutely bornivorous.Indeed, let M ⊂ G be such that we have that sup s∈S |u(s)| < ∞ and since W is absolutely bornivorous, there exists α ∈ K such that S ⊂ αW .But then M ⊂ αD.So, D is an absolutely bornivorous polar subset of G.By our hypothesis, D is a neighborhood of zero in G. Hence, there exist a compact subset Y of X and p ∈ cs(E) This proves that E is polarly absolutely quasi-barrelled.To prove that X is an aw o -space, consider an aw o -bounded subset A of X, x a non-zero element of E and define p(s) = |x (s)|.The set We claim that V absorbs Z. Assume the contrary and let |λ| > 1.There exists a sequence ) with the topology of uniform convergence.Let q ∈ cs(F ), q(g) = g p .Then q is a polar seminorm on F and so the normed space F q is polar.Since (V n ) is not finite on Y , it follows that sup n q(g n ) = ∞.Let π : F → F q be the canonical map and gn = π(g n ). Then a contradiction.This contradiction shows that V absorbs Z and therefore V is an absolutely bornivorous barrel.Thus V is a neighborhood of zero in G. Let K be a compact subset of X and r ∈ cs(E) be such that Then A ⊂ K and so A is relatively compact.This clearly completes the proof.
Proof: The necessity follows from the preceding Theorem.Sufficiency : Let D be a polar absolutely bornivorous subset of G and let H = D o be its polar in G .By Theorem 9.17, the set Then Φ is a strongly bounded subset of E .In fact, let B be a bounded subset of E. The set which proves that Φ is strongly bounded in E .But then Φ is equicontinuous.Hence, there exists p ∈ cs(E) such that Thus W ⊂ D and the result follows.
Corollary 1.8.C c (X) is polarly absolutely quasi-barrelled iff X is an aw o -space.Corollary 1.9.Assume that E = {0}.If E is a bornological space and X an aw o -space, then C c (X, E) is polarly absolutely quasi-barrelled.In particular this happens when E is metrizable.Definition 1.10.A locally convex space E is said to be : (1) polarly ℵ o -barrelled if every w -bounded countable union of equicontinuous subsets of E is equicontinuous.
(3) polarly co-barrelled if every w -null sequence in E is equicontinuous.
(4) If a σ-compact subset A of X is bounding, then A is relatively compact.
Proof: Clearly (1) ⇒ (2) ⇒ (3).(3) ⇒ (4).Let (Y n ) be a sequence of compact subsets of X, such that A = Y n is bounding, and choose a non-zero element u of E .Let p be defined on E by p(s) = |u(s)|.Then u p = 1.By [5, p. 273] there exists It follows that the sequence H = (λ n m n ) is w -null and hence by (3) equicontinuous.Let Y be a compact subset of X and q ∈ cs(E) be such But then A ⊂ Y and so A is relatively compact.Finally, suppose that E is a Fréchet space and let (4) hold.Let (H n ) be a sequence of equicontinuous subsets of the dual space M c (X, E ) of G such that H = H n is wbounded.For each n, the set is compact.Also, the set is bounding by [2, Prop.6.6].By our hypothesis, A is compact.Since E is a Fréchet space, the space F = (C rc (X, E), τ u ) is a Fréchet space whose dual can be identified with M (X, E ).As H is σ(F , F )-bounded, it follows that H is τ u -equicontinuous.Thus, there exists p ∈ cs(E) such that This clearly completes the proof.

Tensor Products
Throughout this section, X, Y will be zero-dimensional Hausdorff topological spaces and E, F Hausdorff locally convex spaces.Let B ou (X) denote the collection of all φ ∈ K X for which |φ| is bounded, upper-semicontinuous and vanishes at infinity.For φ ∈ B ou (X) and p ∈ cs(E), let p φ be the seminorm on C b (X, E) defined by As it is shown in [4], the topology β o is generated by the family of seminorms Let E ⊗ F be the tensor product of E, F equipped with the projective topology.
The bilinear map f g is also bounded.

Theorem 2.1. The space G spanned by the functions
We will finish the proof by showing that there exists h ∈ G such that f − h ∈ W .To this end, we consider the set There exists a y such that (x, y) ∈ D and so φ 1 (x) = 0.The set is open and contains x.Using the compactness of D 2 , we can find a clopen neighborhood W x of x contained in Z x such that p ⊗ q(f (z, y) − f (x, y)) < 1/d 2 for all z ∈ W x and all y ∈ D 2 .In view of the compactness of D 1 , there are Keeping those of the A i which are not empty, we may assume that A k = ∅ for all 1 ≤ k ≤ m.For k = 1, . . ., m, there are pairwise disjoint clopen subsets B k,1 , . . ., B k,n k of Y covering D 2 and y kj ∈ B k,j such that Then h ∈ G.We will prove that for all x ∈ X, y ∈ Y .To see this, we consider the three possible cases.Case I. x / ∈ m k=1 A k .Then h(x, y) = 0. Also (x, y) / ∈ D and thus Case II.x ∈ A k , y ∈ D 2 .There exists j such that y ∈ B k,j .Now Since h(x, y) = f (x k , y kj ), we have Thus f − h ∈ W , which completes the proof.
(2) If p is polar, then, for any u = n i=1 x i ⊗ y i , we have Proof: (1).Let h ∈ F and consider the bilinear map Let ω : E ⊗ F → K be the corresponding linear map.Then x (a j )h(b j ).
Since this holds for all h ∈ F , (1) follows.
and so d ≤ sup j p(a j )q(b j ), which proves that d ≤ p ⊗ q(u).On the other hand, let u = n i=1 x i ⊗ y i and let G be the space spanned by the the set {y 1 , . . ., y n }.Given 0 < t < 1, there exists a basis {w 1 , . . ., w m } of G which is t-orthogonal with respect to the seminorm q.We may write u in the form u = m k=1 z k ⊗ w k .For x ∈ E , |x | ≤ p, we have and so Since 0 < t < 1 was arbitrary, we get that d ≥ p⊗q(u) and so d = p⊗q(u).
Then v is linear and |v| ≤ p φ .Moreover, |v(f )| > θ, which proves that p φ is polar.Theorem 2.4.If E is polar, then there exists a linear homeomorphism The bilinear map be the corresponding continuous linear map.Claim.For each u ∈ G, we have Thus which proves that On the other hand, given 0 < t < 1, there exists a representation u = n k=1 f k ⊗g k of u such that the set {g 1 , . . ., g n } is t-orthogonal with respect to the seminorm q φ 2 .Now Since 0 < t < 1 was arbitrary, we get that (p ⊗ q) φ (ω(u)) ≥ p φ 1 ⊗ q φ 2 (u) and the claim follows.
It is now clear that ω is one-to-one and, for M = ω(G), the map ω : 1.This completes the proof.
For x ∈ E and y ∈ F , we denote by x ⊗ y the unique element of (E ⊗ F ) defined by x ⊗ y (s 1 ⊗ s 2 ) = x (s 1 )y (s 2 ).

Theorem 2.5. Assume that E is polar and let
Proof: Since m 1 is β o -continuous on C b (X, E), there exist φ 1 ∈ B ou (X) and a polar continuous seminorm p on E such that | g dm 1 | ≤ p φ 1 (g) for all g ∈ C b (X, E).Similarly, there exist φ 2 ∈ B ou (Y ) and q ∈ cs(F ) such that | f dm 2 | ≤ q φ 2 (f ) for all f ∈ C b (Y, F ). Consider the bilinear map Then T is continuous since |T (g, f )| ≤ p φ 1 (g) • q φ 2 (f ).Hence the corresponding linear map Let ω be as in the preceding Theorem and M = ω(G).The linear map where A ∈ K(X), B ∈ K(Y ), s 1 ∈ E, s 2 ∈ F , we get that In view of Theorem 2.1, we see that v 1 = ṽ on a β o -dense subspace of C b (X × Y, E ⊗ F ) and hence v 1 = ṽ, which implies that m = µ.This completes the proof.
Proof: Let φ 1 and φ 2 be as in the Theorem.For g It is easy to see that Taking 1 → 0, 2 → 0, we get m p⊗q ≥ m 1 p • m 2 q , which completes the proof.
(2) If the valuation of K is dense or if it is discrete and p(E) ⊂ |K|, then m p (V ) = α.
In view of the compactness of G∩D, there are z i = (x i , y i ) ∈ G∩D, i = 1, . . ., n, such that There are pairwise disjoint clopen rectangles A j × B j , j = 1, . . ., k, such that Since 0 < t < 1 was arbitrary, we get that N m,p⊗q (x, y) ≤ |λ| • N m 1 ,p (x)N m 2 ,q (y).
If the valuation of K is dense or if it is discrete and q(F ) ⊂ |K|, then is in B ou (X × Y ) and, for each locally convex space G, the topology β o on C b (X × Y, G) is generated by the seminorms b j of u and any x ∈ E , with |x | ≤ p, we have Definition 2.6.If m 1 , m 2 , m are as in the preceding Theorem, we will call m the tensor product of m 1 , m 2 and denote it by m 1 ⊗ m 2 .