GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 1, 1999, 33-44

TENSOR PRODUCTS OF NON-ARCHIMEDEAN WEIGHTED SPACES OF CONTINUOUS FUNCTIONS

A. K. KATSARAS AND A. BELOYIANNIS

Abstract. It is shown that the completion of the tensor product of two non-Archimedean weighted spaces of continuous functions is topologically isomorphic to another weighted space. Several applications of this result are given.

1. Introduction

Weighted spaces of continuous functions were introduced in the complex case by L. Nachbin in [1], and in the vector case by J. Prolla in [2]. Many other authors have continued the investigation of such spaces. W. H. Summers has shown in [3] that if X and Y are locally compact topological spaces and U, V Nachbin families on X, Y, respectively, then CU_{0}(X) ⊗ CV_{0}(Y) is topologically isomorphic to a dense subspace of CW_{0}(X × Y), where W = U × V = {u × v : u ∈ U, v ∈ V } and (u × v)(x, y) = u(x)v(y).

The p-adic weighted spaces of continuous functions were introduced by J. P. Q. Carneiro in [4]. Several of the properties of these spaces were studied by the authors in [5] and [6]. In this paper we show that if X, Y are Hausdorff topological spaces, not necessarily locally compact, U, V Nachbin families on X, Y respectively and E a non-Archimedean polar locally convex space, then CU_{0}(X) ⊗ CV_{0}(Y, E) is topologically isomorphic to a dense subspace of CW_{0}(X × Y, E), where W = U × V. We give several applications of this result. We also show that on the space C_{b}(X, E) of all bounded continuous E-valued functions on X, the strict topology defined in [7] is the weighted topology which corresponds to a certain Nachbin family on X.

2. Preliminaries

Throughout this paper, K will stand for a complete non-Archimedean valued field whose valuation is nontrivial. By a seminorm, on a vector

33 space E over K, we mean a non-Archimedean seminorm. Let E be a locally convex space over K. The collection of all continuous seminorms on E will be denoted by cs(E). The algebraic and the topological duals of E will be denoted by E^{∗} and E′, respectively. For a subset B of E, B^{0} denotes its polar subset of E′. A seminorm p on E is called polar if

1991 Mathematics Subject Classification. 46S10.

Key words and phrases. Nachbin families, weighted spaces, non-Archimedean seminorms, tensor product, strict topology.

1072-947X/99/0100-0033$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

p = sup{|f| : f ∈ E^{∗}, |f| ≤ p},

where |f| is defined by |f|(x) = |f(x)|. The space E is called polar if its topology is generated by a family of polar seminorms. If E, F are locally convex spaces over K, then E ⊗ F denotes the projective tensor product of these spaces. By E ˆ⊗F we denote the completion of E ⊗ F. Also, by p ⊗ q we denote the tensor product of the seminorms p and q. For all unexplained terms concerning non-Archimedean spaces we refer to [8].

Next we recall the definition of non-Archimedean weighted spaces. Let X be a Hausdorff topological space and E a locally convex space. The space of all continuous E-valued functions on X is denoted by C(X, E). By C_{b}(X, E) and C_{0}(X, E) we denote the spaces of all members of C(X, E) which are bounded on X or vanish at infinity on X, respectively. In case E = K, we write C(X), C_{b}(X) and C_{0}(X) instead of C(X, K), C_{b}(X, K) and C_{0}(X, K).

A Nachbin family on X is a family V of non-negative upper-semiconti-nuous functions on X such that:

(1) For all v_{1}, v_{2} ∈ V and any a > 0 there exists v ∈ V with v ≥ av_{1}, av_{2} (pointwise) on X.

(2) For every x ∈ X there exists v ∈ V with v(x) > 0.

Let now p ∈ cs(E) and v ∈ V. For an E-valued function f on X, we define

q_{v,p}(f) = ∥f∥_{v,p} = sup{v(x)p(f(x)) : x ∈ X}.

In case f is K-valued, we define

q_{v}(f) = ∥f∥_{v} = sup{v(x)|f(x)| : x ∈ X}.

Also, for an R-valued or K-valued function f on X, we define

∥f∥ = sup{|f(x)| : x ∈ X}.

The weighted space CV (X, E) is defined to be the space of all f in C(X, E) such that q_{v,p}(f) < ∞ for all v ∈ V and all p ∈ cs(E). Note that q_{v,p} is a non-Archimedean seminorm on CV (X, E). We will denote by CV_{0}(X, E) the subspace of CV (X, E) consisting of all f such that the function x ↦ v(x)p(f(x)) vanishes at infinity on X for each v ∈ V and each p ∈ cs(E). On CV (X, E) and on CV_{0}(X, E) we will consider the weighted topology τ_{ν} generated by the seminorms q_{v,p}, v ∈ V, p ∈ cs(E). When E = K, we will simply write CV (X) and CV_{0}(X) instead of CV (X, K) and CV_{0}(X, K).

TENSOR PRODUCTS OF NON-ARCHIMEDEAN WEIGHTED SPACES 35

3. On the Strict Topology

For a locally compact zero dimensional topological space X and a non-Archimedean normed space E, J. Prolla has defined, in [9], the strict topology β on C_{b}(X, E) as the topology defined by the seminorms

f ↦ ∥ϕf∥ = sup{∥ϕ(x)f(x)∥ : x ∈ X},

where ϕ ∈ C_{0}(X). For an arbitrary topological space X and a locally convex space E, the strict topology β_{0} on C_{b}(X, E) was defined in [7]. This is the topology generated by the seminorms

f ↦ ∥ϕf∥_{p} = sup{|ϕ(x)|p(f(x)) : x ∈ X}

where p ∈ cs(E) and ϕ belongs to the family B_{0}(X) of all bounded K-valued functions f on X which vanish at infinity. As shown in [7], β_{0} = β when X is locally compact zero-dimensional. In this section we will show that β_{0} is a weighted topology.

Let X be a Hausdorff topological space and let B_{0u}(X) denote the family of all ϕ ∈ B_{0}(X) for which |ϕ| is upper-semicontinuous.

Lemma 3.1.

(1) If V = |B_{0u}(X)| = {|ϕ| : ϕ ∈ B_{0u}(X)}, then V is a Nachbin family on X.

(2) For each ϕ ∈ B_{0}(X) there exists ψ ∈ B_{0u}(X) such that |ϕ| ≤ |ψ|.

Proof. (1) If ϕ_{1}, ϕ_{2} ∈ B_{0u}(X) and if ϕ is defined on X by

ϕ(x) = \{ \begin{matrix}ϕ_{1}(x) + ϕ_{2}(x) & if |ϕ_{1}(x)| ≠ |ϕ_{2}(x)| \\ ϕ_{1}(x) & otherwise, \end{matrix}

then |ϕ| = max{|ϕ_{1}|, |ϕ_{2}|} and ϕ ∈ B_{0u}(X). It follows now easily that V is a Nachbin family on X.

(2) Let ϕ ∈ B_{0}(X) and choose λ ∈ K, 0 < |λ| < 1. Without loss of generality we may assume that ∥ϕ∥ < |λ|. There exists an increasing sequence (D_{n}) of compact subsets of X such that {x ∈ X : |ϕ(x)| > |λ|^{n}} ⊆ D_{n}. Let ϕ_{n} denote the K-characteristic function of D_{n}. For each x ∈ X, the series ∑_{n=1}^{∞} λ^{n}ϕ_{n}(x) converges in K. Define ψ on X by

ψ(x) = ∑_{n=1}^{∞} λ^{n}ϕ_{n}(x).

If x ∈ D_{n} \ D_{n−1}, then |ψ(x)| = |λ|^{n}. Given ϵ > 0, choose n such that |λ|^{n} < ϵ. Now {x ∈ X : |ψ(x)| > ϵ} ⊆ D_{n} and so ψ ∈ B_{0}(X). Also, for each ϵ > 0, the set A = {x : |ψ(x)| < ϵ} is open. Indeed, if |λ| < ϵ, then A = X. Assume ϵ ≤ |λ| and let κ be such that |λ|^{κ+1} < ϵ ≤ |λ|^{κ}. If x_{0} ∈ A, then x_{0} ∉ D_{κ}. Also, for x ∉ D_{κ}, we have |ψ(x)| ≤ |λ|^{κ+1} < ϵ and so x ∈ A. Thus A = X \ D_{κ}, which shows that A is open. Finally, |λϕ| ≤ |ψ|. Indeed, let ϕ(x) ≠ 0. If x ∈ D_{1}, then |ψ(x)| = |λ| ≥ |λϕ(x)|. If x ∈ D_{n+1} \ D_{n}, then |ϕ(x)| ≤ |λ|^{n} and so |ψ(x)| = |λ|^{n+1} ≥ |λϕ(x)|.

Theorem 3.2. If V is as in the preceding Lemma, then

CV (X, E) = CV_{0}(X, E) = C_{b}(X, E) (algebraically)

and the weighted topology on CV (X, E) coincides with the strict topology β_{0} on C_{b}(X, E).

Proof. It is clear that C_{b}(X, E) ⊆ CV_{0}(X, E). On the other hand, assume that some f ∈ CV (X, E) is not bounded. Then, for |λ| > 1, there exist p ∈ cs(E) and a sequence (x_{n}) of distinct elements of X such that p(f(x_{n})) > |λ|^{2n} for all n. Let ϕ_{n} be the K-characteristic function of the set {x_{1}, …, x_{n}}. As in the proof of the preceding Lemma, we get that the function ϕ = ∑_{n=1}^{∞} λ^{−n}ϕ_{n} is in B_{0u}(X) and |ϕ(x_{n})| = | ∑_{κ≥n} λ^{−κ}ϕ_{κ}(x_{n})| = |λ|^{−n}. Thus sup_{n} |ϕ(x_{n})|p(f(x_{n})) = ∞ contradicts the fact that f ∈ CV (X, E). This proves the first part. The second part follows from (2) of the preceding Lemma.

4. Tensor Products of Weighted Spaces

Let X, Y be Hausdorff topological spaces and let U, V be Nachbin families on X, Y respectively. Set W = U × V = {u × v : u ∈ U, v ∈ V } where u × v is defined on X × Y by (u × v)(x, y) = u(x)v(y). It is easy to see that W is a Nachbin family on X × Y. In the complex case, Summers has shown in [3] that, for locally compact X, Y, CU_{0}(X) ⊗ CV_{0}(Y) is topologically isomorphic to a dense subspace of CW_{0}(X ×Y). The following is an analogous result in our case. Note that we do not assume that X, Y are locally compact.

Theorem 4.1. Let U, V, W be as above and let E be a Hausdorff locally convex space over K. Then:

(1) CU_{0}(X) ⊗ CV_{0}(Y, E) is topologically isomorphic to a subspace G of CW_{0}(X × Y, E);

(2) if X is zero-dimensional and E a polar space, then G is a dense subspace of CW_{0}(X × Y, E).

Proof. (1) Let

B : CU_{0}(X) × CV_{0}(Y, E) ↦ CW_{0}(X × Y, E),
B(ϕ, f) = ϕ × f, (ϕ × f)(x, y) = ϕ(x)f(y).

Then B is bilinear. Let

T = B̃ : CU_{0}(X) ⊗ CV_{0}(Y, E) ↦ CW_{0}(X × Y, E)

be the corresponding linear map. Then T is one-to-one. Indeed, assume that for some h = ∑_{1}^{n} ϕ_{κ} ⊗ f_{κ} we have T (h) = 0. We claim that h = 0. We prove it by induction on n. This is clearly true if n = 1. Assume that it is true for n − 1. If some ϕ_{κ} ≠ 0, say ϕ_{n} ≠ 0, then f_{n} is a linear combination of f_{1}, …, f_{n−1}, i.e., f_{n} = ∑_{κ=1}^{n−1} λ_{κ}f_{κ}. Thus

0 = ∑_{1}^{n} ϕ_{κ} × f_{κ} = ∑_{1}^{n−1} ϕ_{κ} × f_{κ} + ∑_{1}^{n−1} λ_{κ}(ϕ_{n} × f_{κ}) = ∑_{1}^{n−1}(ϕ_{κ} + λ_{κ}ϕ_{n}) × f_{κ}.

By our inductive hypothesis, we have

0 = ∑_{1}^{n−1}(ϕ_{κ} + λ_{κ}ϕ_{n}) ⊗ f_{κ} = ∑_{1}^{n−1} ϕ_{κ} ⊗ f_{κ} + ∑_{1}^{n−1} λ_{κ}ϕ_{n} ⊗ f_{κ} =

= ∑_{1}^{n−1} ϕ_{κ} ⊗ f_{κ} + ϕ_{n} ⊗ ( ∑_{1}^{n−1} λ_{κ}f_{κ}) = ∑_{1}^{n} ϕ_{κ} ⊗ f_{κ}.

This proves that T is one-to-one. Also, if M = CU_{0}(X) ⊗ CV_{0}(Y, E) and G = T (M), then T is a topological isomorphism from M onto G. Indeed, let h ∈ M, u ∈ U, v ∈ V, w = u × v, p ∈ cs(E). For any representation h = ∑_{1}^{n} ϕ_{κ} ⊗ f_{κ} of h we have

∥T h∥_{w,p} = sup_{x,y} u(x)v(y) p( ∑_{1}^{n} ϕ_{κ}(x)f_{κ}(y)) ≤

≤ max_{κ} [( sup_{x} u(x)|ϕ_{κ}(x)|) · ( sup_{y} v(y)p(f_{κ}(y)))] = max_{κ} ∥ϕ_{κ}∥_{u}∥f_{κ}∥_{v,p}.

Thus ∥T h∥_{w,p} ≤ (∥ · ∥_{u} ⊗ ∥ · ∥_{v,p}) (h). On the other hand, given 0 < t < 1, there exists a representation h = ∑_{κ=1}^{m} ϕ_{κ} ⊗ f_{κ} of h such that {f_{1}, …, f_{m}} is t-orthogonal with respect to the seminorm ∥ · ∥_{v,p}. Now, for any x ∈ X,

∥ ∑_{κ=1}^{m} ϕ_{κ}(x)f_{κ}∥_{v,p} ≥ t max_{κ} |ϕ_{κ}(x)|∥f_{κ}∥_{v,p}

and so

∥T h∥_{w,p} = sup_{x} [∥ ∑_{1}^{m} ϕ_{κ}(x)f_{κ}∥_{v,p}]u(x) ≥ t max_{κ} sup_{x} |ϕ_{κ}(x)|∥f_{κ}∥_{v,p}u(x) =

= t max_{κ} ∥ϕ_{κ}∥_{u} ∥f_{κ}∥_{v,p} ≥ t (∥ · ∥_{u} ⊗ ∥ · ∥_{v,p}) (h).

It follows that ∥T h∥_{w,p} = (∥ · ∥_{u} ⊗ ∥ · ∥_{v,p}) (h) and so T : M ↦ G is a topological isomorphism.

(2) Assume that E is polar and X zero-dimensional.

Let f ∈ CW_{0}(X × Y, E), u ∈ U, v ∈ V, w = u × v, ϵ > 0 and p ∈ cs(E), where p is polar. The set D = {(x, y) : u(x)v(y) p(f(x, y)) ≥ ϵ} is compact in X × Y. If D_{1}, D_{2} are the projections of D on X, Y respectively, then D ⊆ D_{1} × D_{2}. Let d > sup_{x∈D_{1}} u(x), sup_{y∈D_{2}} v(y).

The set Ω = {x ∈ X : u(x) < d} is open in X and contains D_{1}. Since X is zero-dimensional, there exists a clopen subset A of X with D_{1} ⊆ A ⊆ Ω. For each x ∈ D_{1} there exists y ∈ Y with (x, y) ∈ D and so u(x) > 0. Also, for x_{0} ∈ X, the map y ↦ f(x_{0}, y) is in CV_{0}(Y, E). Indeed, there exists u_{1} ∈ U with u_{1}(x_{0}) ≠ 0. Let v_{1} ∈ V, ϵ_{1} > 0 and q ∈ cs(E). We want to show that the set B = {y ∈ Y : v_{1}(y) q(f(x_{0}, y)) ≥ ϵ_{1}} is compact. The set B_{1} = {(x, y) : u_{1}(x)v_{1}(y)q(f(x, y)) ≥ ϵ_{1}u_{1}(x_{0})} is compact. If y ∈ B, then (x_{0}, y) ∈ B_{1} and so B is contained in the projection of B_{1} in Y. Since B is closed, it follows that B is compact. This proves that the map y ↦ f(x_{0}, y) is in CV_{0}(Y, E).

Also, for each y_{0} ∈ Y and each x′ ∈ E′, the function x ↦ x′(f(x, y_{0})) is in CU_{0}(X). Indeed, the seminorm q(x) = |x′(x)| is continuous on E. Choose v_{1} ∈ V with v_{1}(y_{0}) ≠ 0. For u_{1} ∈ U, let H = {x : u_{1}(x)q(f(x, y_{0})) ≥ ϵ_{1}}. Then, H is contained in the projection on X of the compact set B_{2} = {(x, y) : u_{1}(x)v_{1}(y) q(f(x, y)) ≥ ϵ_{1}v_{1}(y_{0})} and so H is compact, which proves that the function x ↦ x′(f(x, y_{0})) is in CU_{0}(X).

Let now x∈D_{1}. There exists y_{0} ∈ Y with (x, y_{0})∈D. Since p(f(x, y_{0}))>0 and p is polar, there exists x′ ∈ E′ with x′(f(x, y_{0})) ≠ 0. Since the function z ↦ x′(f(z, y_{0})) is in CU_{0}(X), it is clear that there exists ϕ_{x} ∈ CU_{0}(X) with ϕ_{x}(x) = 1. By the compactness of D_{2}, there exists a clopen neighborhood A_{x} and 0 < ϵ_{x} < 1, with

d^{2} · ϵ_{x} · sup_{y∈D_{2}} p(f(x, y)) < ϵ,

such that

A_{x} ⊆ A ∩ {z : |ϕ_{x}(z) − 1| < ϵ_{x}} ∩ {z : u(z) < 2u(x)}

and p(f(z, y) − f(x, y)) < ϵ/d^{2} for all z ∈ A_{x} and all y ∈ D_{2}. In view of the

compactness of D_{1}, there are x_{1}, …, x_{m} in D_{1} such that D_{1} ⊆ ⋃_{1}^{m}A_{x_{i}}.

Let A_{1} = A_{x_{1}}, A_{κ+1} = A_{x_{κ+1}} \ ( ⋃_{1}^{κ} A_{x_{i}}) for κ = 1, …, m − 1.

Set ϕ_{κ} = ϕ_{x_{κ}} · X_{A_{κ}}, f_{κ} = f(x_{κ}, ·) ∈ CV_{0}(Y, E), where X_{A_{κ}} is the Kcharacteristic function of A_{κ}. Then h = ∑_{1}^{m} ϕ_{κ} × f_{κ} ∈ G. Moreover, for all x ∈ X and y ∈ Y, we have

u(x)v(y) p(f(x, y) − h(x, y)) ≤ 2ϵ.  (∗)

To show (∗) we consider three possible cases.

Case I: x ∉ ⋃_{1}^{m} A_{κ}.

In this case, we have h(x, y) = 0, (x, y) ∉ D and u(x)v(y) p(f(x, y)) < ϵ. Case II: x ∈ A_{κ} and y ∈ D_{2}.

Then

f(x, y) − h(x, y) = f(x, y) − ϕ_{κ}(x)f_{κ}(y) =
= [f(x, y) − f(x_{κ}, y)] + f(x_{κ}, y)(1 − ϕ_{κ}(x)).

Since

u(x)v(y) p (f(x, y) − f(x_{κ}, y)) < d^{2} · ϵ/d^{2} = ϵ

and

u(x)v(y)|1 − ϕ_{κ}(x)| p(f(x_{κ}, y)) ≤ d^{2} · ϵ_{x_{κ}} · p(f(x_{κ}, y)) < ϵ,

we have that (∗) holds.

Case III: x ∈ A_{κ}, y ∉ D_{2}.

In this case we have that (x, y) ∉ D and so u(x)v(y)p(f(x, y)) < ϵ. Also, since x ∈ A_{κ} ⊆ A_{x_{κ}}, we have ϕ_{κ}(x) = ϕ_{x_{κ}}(x) and |ϕ_{x_{κ}}(x) − 1| < 1, which implies that |ϕ_{x_{κ}}(x)| = 1. Thus

u(x)v(y)|ϕ_{κ}(x)|p(f(x_{κ}, y)) ≤ 2u(x_{κ})v(y)p(f(x_{κ}, y)) < 2ϵ,

since (x_{κ}, y) ∉ D. Thus (∗) holds in all cases and so ∥f − h∥_{w,p} ≤ 2ϵ.

Remark 4.2. Looking at the proof of (2) in the preceding Theorem, we see that instead of the hypothesis that E is polar we may just assume that E′ separates the points of E, i.e., for each s ≠ 0 in E there exists x′ ∈ E′ with x′(s) ≠ 0. Of course polar spaces have this property.

Taking as V the family of all constant positive functions on X, we get that CV_{0}(X, E) coincides with C_{0}(X, E) with the topology τ_{u} of uniform convergence.

Lemma 4.3. Considering on both C_{0}(X, E) and C_{0}(X, Ê) the topology τ_{u} of uniform convergence, we have that C_{0}(X, Ê) is the completion of C_{0}(X, E).

Proof. It is easy to see that C_{0}(X, Ê) is complete. Let f ∈ C_{0}(X, Ê) and p ∈ cs(E). We will denote also by p the unique continuous extension of p to all of Ê.

The set Z = {x ∈ X : p(f(x)) ≥ 1} is clopen and compact. There are x_{1}, …, x_{n} in Z such that the sets

Z_{κ} = {x ∈ X : p(f(x) − f(x_{κ}))_{≤}1} , κ = 1, …, n,

are pairwise disjoint and cover Z. For each κ, choose s_{κ} ∈ E with p(s_{κ} − f(x_{κ})) < 1. Set

h = ∑_{1}^{n} X_{A_{κ}}s_{κ} ∈ C_{0}(X, E),

where A_{κ} = Z_{κ} ∩ Z. Note that the sets A_{1}, …, A_{n} are clopen and compact and their union is Z. Since ∥f − h∥_{p} ≤ 1, the result follows.

Combining Theorem 1 with Lemma 2, we get as a corollary the following

Theorem 4.4. Let X, Y be Hausdorff topological spaces and E a Hausdorff locally convex space. Then:

(1) C_{0}(X) ⊗ C_{0}(Y, E) is topologically isomorphic to a subspace of C_{0}(X × Y, E);

(2) if X is zero-dimensional and E′ separates the points of E (e.g. if E is polar), then

C_{0}(X) ˆ⊗C_{0}(Y, E) ≅ C_{0}(X × Y, Ê).

Lemma 4.5. Let X, Y be Hausdorff topological spaces, U = |B_{0u}(X)|, V = |B_{0u}(Y)|, W = U ×V, W_{1} = |B_{0u}(X ×Y)|. Then, the Nachbin families W and W_{1} are equivalent.

Proof. Clearly, W ⊆ W_{1}. On the other hand, let ϕ ∈ B_{0u}(X × Y) and λ ∈ K, μ ∈ K with |μ| > 1, |λ| ≥ |μ|^{2}. Without loss of generality, we may assume that ∥ϕ∥ < |λ|^{−1}. For each positive integer n, the set

D_{n} = {(x, y) : |ϕ(x, y)| ≥ |λ|^{−n}}

is compact. Let A_{n}, B_{n} be the projections of D_{n} on X, Y, respectively. Set

ϕ_{1} = ∑_{n=1}^{∞} μ^{−n}X_{A_{n}} , ϕ_{2} = ∑_{n=1}^{∞} μ^{−n}X_{B_{n}}.

Since (A_{n}), (B_{n}) are increasing sequences of compact sets, we get (as in the proof of Lemma 1) that ϕ_{1} ∈ B_{0u}(X) and ϕ_{2} ∈ B_{0u}(Y). Moreover, |ϕ| ≤ |λ|(|ϕ_{1}| × |ϕ_{2}|). Indeed, let (x_{0}, y_{0}) ∈ X × Y with ϕ(x_{0}, y_{0}) ≠ 0, and let n be the smallest of all integers κ with (x_{0}, y_{0}) ∈ D_{κ}. If m is the smallest integer κ with x_{0} ∈ A_{κ}, then m ≤ n and |ϕ_{1}(x_{0})| = |μ|^{−m} ≥ |μ|^{−n}. Similarly, |ϕ_{2}(y_{0})| ≥ |μ|^{−n} and so

|ϕ_{1}(x_{0})ϕ_{2}(y_{0})| ≥ |μ|^{−2n} ≥ |λ|^{−n}.

Since (x_{0}, y_{0}) ∉ D_{n−1}, we have that

|ϕ(x_{0}, y_{0})| < |λ|^{−(n−1)} ≤ |λ||ϕ_{1}(x_{0})ϕ_{2}(y_{0})|.

This clearly completes the proof.

Combining the preceding Lemma with Theorems 3.2 and 4.1, we get

Theorem 4.6. Let X, Y be Hausdorff topological spaces and E a Hausdorff locally convex space. Then:

(1) (C_{b}(X), β_{0}) ⊗ (C_{b}(Y, E), β_{0}) is topologically isomorphic to a subspace G of (C_{b}(X × Y, E), β_{0}) = M.

(2) If X is zero-dimensional and E′ separates the points of E, then G is a dense subspace of M.

Let X, Y be Hausdorff topological spaces, U the Nachbin family of all positive multiples of the R-characteristic functions of the compact subsets of X, V = |B_{0u}(Y)| and W = U × V. Let f ∈ E^{X×Y} be such that the restriction f|_{D} to each compact subset D of X × Y is continuous.

Consider the following properties of f:

(1) For each compact subset D_{1} of X, the restriction of f to D_{1} × Y is bounded.

(2) For any u ∈ U, v ∈ V, w = u × v, p ∈ cs(E), the function w · (p ∘ f) vanishes at infinity on X × Y.

(3) ∥f∥_{w,p} < ∞ for any w = u × v ∈ W and any p ∈ cs(E). Then (1), (2), (3) are equivalent. Indeed, it is easy to see that (1) ⇒ (2) ⇒ (3). To prove that (3) ⇒ (1), assume that there exist a compact subset D_{1} of X and p ∈ cs(E) such that

sup{p(f(x, y)) : x ∈ D_{1}, y ∈ Y } = ∞.

Let |λ| > 1 and choose a sequence (x_{n}) in D_{1} and a sequence (y_{n}) of distinct elements of Y such that p(f(x_{n}, y_{n})) > |λ|^{2n}. Let w_{n} be the K-characteristic function of {y_{1}, …, y_{n}} and consider the function ϕ = ∑_{n=1}^{∞} λ^{−n}w_{n}. Then v = |ϕ| ∈ V. If u is the R-characteristic function of D_{1}, then w = u×v ∈ W and

u(x_{n})v(y_{n})p(f(x_{n}, y_{n})) = |λ|^{−n}p(f(x_{n}, y_{n})) ≥ |λ|^{n}

and so ∥f∥_{w,p} = ∞, a contradiction. Thus (1),(2),(3) are equivalent.

Let now U, V, W be as above and denote by CW_{κ}(X × Y, E) the vector space of all f ∈ E^{X×Y} such that:

(a) f|_{D×Y} is continuous for each compact subset D of X.

(b) ∥f∥_{w,p} < ∞ for each w ∈ W and each p ∈ cs(E).

If we consider on CW_{κ}(X × Y, E) the weighted topology τ_{w} generated by the seminorms ∥ · ∥_{w,p}, w ∈ W, p ∈ cs(E), we have

Theorem 4.7. Let X, Y be zero-dimensional Hausdorff topological spaces and E a Hausdorff locally convex space. If τ_{c} is the topology of compact convergence, then:

(1) the map

ω : (C(X), τ_{c}) ⊗ (C_{b}(Y, E), β_{0}) ↦ CW_{κ}(X × Y, E), f ⊗ g ↦ f × g,

is a topological isomorphism onto a dense subspace G of CW_{κ}(X × Y, E); (2) if Y is locally compact, then

(C(X), τ_{c}) ˆ⊗(C_{b}(Y, E), β_{0}) ≅ CW_{κ}(X × Y, Ê).

Proof. The mapping ω is a topological isomorphism onto G by Theorem 4.1, since CW_{0}(X × Y, E) is a topological subspace of CW_{κ}(X × Y, E). To prove that G is dense, let f ∈ CW_{κ}(X × Y, E), w = u × v ∈ W and p ∈ cs(E).

We may assume that u is the R-characteristic function of a compact subset D_{1} of X. Given ϵ > 0, let D = {(x, y) : x ∈ D_{1}, v(y)p(f(x, y)) ≥ ϵ}. If D_{2} is the projection of D on Y, then D_{2} is compact, since D is compact, and D ⊆ D_{1} × D_{2}. The restriction h of f to D_{1} × D_{2} is continuous. Let ϵ_{2} > 0 with ϵ_{2}∥v∥ < ϵ. There are (x_{κ}, y_{κ}) ∈ D_{1} × D_{2}, κ = 1, …, n, such that the sets A_{κ} = {s ∈ E : p(s − f(x_{κ}, y_{κ})) ≤ ϵ_{2}} are pairwise disjoint and cover h(D_{1} × D_{2}). Set B_{κ} = h^{−1}(A_{κ}). Clearly, B_{κ} is compact and D_{1} × D_{2} = ⋃_{κ} B_{κ}.

It is easy to see that if C, C_{1}, …, C_{n} are clopen in X and F, F_{1}, …, F_{n} clopen in Y, then the set

C × F \ ( ⋃_{κ=1}^{n} C_{κ} × F_{κ})

is a finite disjoint union of sets of the form Z_{1} × Z_{2}, with Z_{1} clopen in X and Z_{2} clopen in Y.

There are pairwise disjoint sets O_{1} …, O_{n} in X × Y with B_{κ} ⊆ O_{κ}. For (x, y) ∈ B_{κ} there are clopen neighbourhoods M_{x}, D_{y} of x, y respectively such that M_{x} × D_{y} ⊆ O_{κ} and p(f(x, y) − f(a, b)) ≤ ϵ_{2} for all a ∈ M_{x} ∩ D_{1} and b ∈ D_{y}. In view of the compactness of B_{κ}, there are clopen sets A_{κ1}, …, A_{κm_{κ}} in X and clopen sets D_{κ1}, …, D_{κm_{κ}} in Y such that the sets A_{κj} × D_{κj}, j = 1, …, m_{κ}, are pairwise disjoint, cover B_{κ}, are contained in O_{κ} and p(f(x, y) − f(a, b)) ≤ ϵ_{2} if (x, y) and (a, b) are in (A_{κj} ∩ D_{1}) × D_{κj}.

Choose (x_{κj}, y_{κj}) ∈ (A_{κj} ∩ D_{1}) × D_{κj} and set

g = ∑_{κ=1}^{n} ( ∑_{j=1}^{m_{κ}} X_{C_{κj}} × (X_{F_{κj}}f(x_{κ}, y_{κ})))

is in G. Moreover, ∥f − g∥_{w,p} ≤ ϵ. Indeed, let x ∈ D_{1}, y ∈ Y.

Case I: (x, y) ∈ A_{κj} × B_{κj}.

Then g(x, y) = f(x_{κ}, y_{κ}) and so p(f(x, y) − g(x, y)) ≤ ϵ_{2}, which implies that

v(y)p(f(x, y) − g(x, y)) ≤ ∥v∥ϵ_{2} < ϵ.

Case II: (x, y) ∉ ⋃_{κ,j} A_{κj} × B_{κj}.

Then g(x, y) = 0 and (x, y) ∉ D and so

w(x, y)p(f(x, y) − g(x, y)) ≤ v(y)p(f(x, y)) < ϵ.

This proves the first part of the theorem.

(2) To prove the second part, we show first that CW_{κ}(X × Y, Ê) is complete. To this end, let (f_{α}) be a Cauchy net in CW_{κ}(X × Y, Ê).

Given (x_{0}, y_{0}) ∈ X × Y, there exist u ∈ U, v ∈ V with u(x_{0}) > 0, v(y_{0}) > 0. Using this, we get that the net (f_{α}(x_{0}, y_{0})) is Cauchy and hence convergent in Ê. Define

f : X × Y ↦ Ê, f(x, y) = lim f_{α}(x, y).

(i) For each compact subset D_{1} of X, f|_{D_{1}×Y} is continuous. Indeed, let x_{0} ∈ D_{1} and y_{0} ∈ Y. There exists a compact clopen neighbourhood W of y_{0} ∈ Y.

If u, v are the R-characteristic functions of D_{1}, W, respectively, then w = u × v ∈ W and

∥f_{α} − f_{β}∥_{w,p} = sup{p(f_{α}(x, y) − f_{β}(x, y)) : x ∈ D_{1}, y ∈ W }.

It follows that f_{α} → f uniformly on D_{1} × W. Since D_{1} × W is open in D_{1} × Y and (x_{0}, y_{0}) ∈ D_{1} × W it follows that f is continuous at (x_{0}, y_{0}) on D_{1} × Y.

(ii) If w = u×v ∈ W, then ∥f∥_{w,p} < ∞ for each p ∈ cs(E). Indeed, there exists α_{0} such that ∥f_{α_{0}} − f_{α}∥_{w,p} ≤ 1, for all α ⪰ α_{0}, which implies that

∥f_{α_{0}} − f∥_{w,p} ≤ 1 and so ∥f∥_{w,p} ≤ max{1, ∥f_{α_{0}}∥_{w,p}} < ∞.

It follows from the above that f ∈ CW_{κ}(X × Y, Ê) and f_{α} → f in the topology τ_{w}. To finish the proof, it suffices to show that CW_{κ}(X × Y, E) is dense in CW_{κ}(X × Y, Ê). So, let f ∈ CW_{κ}(X × Y, Ê), w = u × v ∈ W and p ∈ cs(E). As in the proof of the first part, there are clopen subsets A_{1}, …, A_{n} of X, clopen subsets B_{1}, …, B_{n} of Y and (x_{κ}, y_{κ}) in X ×Y such that the sets A_{κ} ×B_{κ}, κ = 1, …, n, are pairwise disjoint and ∥f −g∥_{w,p} ≤ 1, where

g = ∑_{κ=1}^{n} X_{A_{κ}} × (X_{B_{κ}}f(x_{κ}, y_{κ})) .

Since w is bounded, we have that ∥w∥ = d < ∞. For each κ, choose s_{κ} ∈ E such that p(s_{κ} − f(x_{κ}, y_{κ})) < 1/d. Now

h = ∑_{κ=1}^{n} X_{A_{κ}} × (X_{B_{κ}}s_{κ}) ∈ G.

If (x, y) ∈ A_{κ} × B_{κ}, then g(x, y) = f(x_{κ}, y_{κ}), h(x, y) = s_{κ}, and so

w(x, y)p(f(x, y) − h(x, y)) ≤
≤ max{w(x, y)p(f(x, y) − g(x, y)), w(x, y)p(f(x_{κ}, y_{κ}) − s_{κ})} ≤ 1.

Thus ∥f − h∥_{w,p} ≤ 1 and the result clearly follows.

Let C_{κ,0}(X ×Y, E) denote the space of all E-valued functions f on X ×Y such that f|_{D_{1}×Y} ∈ C_{0}(D_{1} × Y, E) for each compact subset D_{1} of X. If we consider on C_{κ,0}(X × Y, E) the locally convex topology generated by the seminorms ∥f∥_{D_{1},p} = sup{p(f(x, y) : x ∈ D_{1}, y ∈ Y }, where p ∈ cs(E) and D_{1} is a compact subset of X, then we have

Theorem 4.8. Let X, Y be zero-dimensional Hausdorff topological spaces, where Y is locally compact, and let E be a Hausdorff complete locally convex space. Then

(C(X), τ_{c}) ˆ⊗ (C_{0}(Y, E), τ_{u}) ≅ C_{κ,0}(X × Y, E).

Proof. The proof is analogous to the one of the preceding theorem, using an additional fact that the clopen compact subsets of Y form the base for the open subsets of Y.

References

1. L. Nachbin, Elements of approximation theory. Van Nostrand Math. Studies, 14, Princeton, Jersey, 1967; reprinted in 1976 by Kreiger, Melbourne; initial publications: Notes de Matematica, No. 33, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, 1965.

2. J. B. Prolla, Weighted spaces of vector-valued continuous functions. Ann. Mat. Pura Appl. 89(1971), No. 4, 145–158.

3. W. H. Summers, A representation theorem for biequicontinuous completed tensor products of weighted space. Trans. Amer. Math. Soc. 146(1969), 121–131.

4. J. P. Q. Carneiro, Non-Archimedean weighted approximation. (Portuguese) An. Acad. Brasil. Ciènc. 50(1)(1978), 1–34.

5. A. K. Katsaras and A. Beloyiannis, Non-Archimedean weighted spaces of continuous functions. Rendi. Mat. Appl. 16(1996), 545–562.

6. A. K. Katsaras and A. Beloyiannis, On non-Archimedean weighted spaces of continuous functions. Proc. Fourth Inter. Conference on p-adic Analysis (Nijmegen, The Netherlands), 237–252, Marcel Dekker, 1997.

7. A. K. Katsaras, The Strict topology in non-Archimedean vector-valued function spaces. Proc. Konink. Nederl. Akad. Wetensch. A 87(2)(1984), 189–201.

8. W. H. Schikhoff, Locally convex spaces over non-spherically complete fields I,II. Bull. Soc. Math. Belg., Ser B 38(1986), 187–224.

9. J. B. Prolla, Approximation of vector-valued functions. North Holland Publ. Co., Amsterdam, New York, Oxford, 1977.

(Received 26.11.1996)

Authors’ address:

Department of Mathematics

University of Ioannina

P.O. Box 1186, 451 10 Ioannina

Greece