About the density property in the space of continuous maps vanishing at infinity

The conditions when C0(X)⊗Y is dense in C0(X ,Y ) in the compact-open topology on C0(X ,Y ) are given. This result is used for describing the properties of topological Segal algebras.


INTRODUCTION
Let K denote either the field R of real numbers or the field C of complex numbers, X a topological space and Y a topological linear space over K (shortly, a topological linear space), C(X,Y ) the set of all continuous maps from X to Y , and C 0 (X,Y ) the subset of all such f ∈ C(X,Y ) that vanish at infinity. In case we want to specify the topology of a topological space X, we write instead of X a pair (X, τ X ), where τ X denotes the topology of X.
In [1], some results about Segal algebras were obtained, where one of the conditions that had to be fulfilled was that the set C 0 (X, K) ⊗ B (the full definition of this set will be given further in this paper) had to be dense in C 0 (X, B) (in the compact-open topology) for a topological algebra B. In [2], a result (Theorem 1 on page 27) is given describing the density of a subset of C(X, K) ⊗Y in C(X,Y ) for a Tikhonov space X and topological linear Hausdorff space Y over K (again in the compact-open topology). In [10], some similar results (Theorem 1 on page 98, Corollaries 1 and 2 on page 99) are presented for a compact Hausdorff space X and a topological linear space Y . It appeares that some of the ideas of [2] were such that they could be modified in order to obtain the density needed in our case. The present paper gives some sufficient conditions on a topological space X and a topological linear space Y under which the set C 0 (X, K) ⊗ Y is dense in C 0 (X,Y ) in the compact-open topology. The obtained results will be applied to the results of [1] at the end of the paper.

PRELIMINARY DEFINITIONS AND RESULTS
Let Y be a topological linear space and K a subset of Y . Definition 2.1. A map L : K → Y is said to be finite-dimensional 1 if there exist a positive integer n and an n-dimensional subspace Z of Y such that L(K) ⊆ Z. Moreover, a finite-dimensional map L : K → Y , which can be represented in a form L(y) = λ 1 (y)e 1 + · · · + λ n (y)e n for every y ∈ K, where {e 1  Remark 2.4. The term 'nonlinear approximation property' was suggested for that class of topological linear spaces already by Waelbroeck in [12] in 1972.
It is easy to see that every topological linear space that has the approximation property has also the nonlinear approximation property, and that every topological space that has the nonlinear approximation property is Klee admissible.
In [5], p. 826, the authors claim that every locally convex space is Klee admissible and pose an open problem to find out whether every topological linear space is Klee admissible.
In the proofs of the present paper we need some 'stronger' versions of the Klee admissibility and nonlinear approximation property. Note that the condition L(θ Y ) = θ Y gives for a finite-dimensional map L : Y → Y , which can be written as L(y) = λ 1 (y)e 1 + · · · + λ n (y)e n , that λ 1 (θ Y ) = · · · = λ n (y) = θ Y . Definition 2.6. Let X,Y be topological spaces. It is said that a map f : X → Y vanishes at infinity if for every neighbourhood U of zero in Y there exists a compact set K ⊂ X such that f (x) ∈ U for every x ∈ X \ K.
The following lemmas and corollary will be used in the proofs later. Lemma 2.7. Let X be a topological space and Y a topological linear space. If f ∈ C 0 (X,Y ) and Now let us discuss the property of continuous coordinate functions. Proof. By Theorem 1 from [6], p. 141, we know that Z is isomorphic to K n via the homeomorphism for all y ∈ Y . Note that the maps p i : Proof. Let Y be a Hausdorff topological linear space and L : Y → Y an arbitrary continuous finitedimensional map. Then there exist a positive integer n, an n-dimensional subspace Z of Y , and basis {e 1 , . . . , e n } of Z such that L(y) = λ 1 (y)e 1 + · · · + λ n (y)e n for every y ∈ Y . But then, by Lemma 2.8, L has continuous coordinate functions. Since L : Y → Y was an arbitrary continuous finite-dimensional map, all continuous finite-dimensional maps L : Y → Y have continuous coordinate functions. Therefore, Y has continuous coordinate functions.
Definition 2.10. A topological space X is a completely regular Hausdorff space if for every closed subset Z of X and every x ∈ X \ Z there is a continuous map f : X → [0, 1] such that f (x) = 0 and f (z) = 1 for every z ∈ Z.
Lemma 2.11. Let Y be a topological linear Hausdorff space, K a compact subset of Y , and f ∈ C(K, K).

every continuous K-valued map on a compact subset K of Y has a continuous extension to the whole space Y .
Proof. Every Hausdorff topological linear space is a Hausdorff topological group, which is a completely regular (Hausdorff) space by Theorem 5 in [7], p. 49. Every compact set in a completely regular (Hausdorff) space is C-embedded (which means that every continuous real-valued function on a compact subset of a completely regular space can be extended to a continuous real-valued function on the whole space) by 3.11 (c) in [4], p. 43.
Hence, every continuous real-valued map on a compact subset K of a Hausdorff topological linear space Y has a real-valued extension to the whole space Y and the case for K = R is proved. Let are defined as f r (y) = a, f i (y) = b for every y ∈ K with f (y) = a + bi. Now, by the first part of the proof, there exist continuous extensions

RESULTS CONNECTED WITH THE DENSITY PROPERTY
Let X be a locally compact Hausdorff space and (Y, τ Y ) a topological linear space. Consider the algebra (C 0 (X,Y ), c Y ) of all continuous maps f : X → Y vanishing at infinity equippped with the compact-open topology c Y , where the subbase of the topology c Y on C 0 (X,Y ) consists of all sets of the form Next, we will give three results similar to Theorem 1 (β ) from [2], p. 27.  If Y is a Hausdorff space, then it has continuous coordinate functions by Corollary 2.9. Since in both cases Y has continuous coordinate functions, the map L has the form L(y) = λ 1 (y)e 1 + · · · + λ n (y)e n and the maps λ i : Y → K are continuous with λ i (θ Y ) = 0 for all i ∈ {1, . . . , n}.
for each x ∈ K. Hence, for every f ∈ C 0 (X,Y ) and every neighbourhood O( f ) of f in C 0 (X,Y ) there exist integer n > 0, g 1 , . . . , g n ∈ C 0 (X, K), and e 1 , . . . , e n ∈ Y such that Therefore, C 0 (X, K) ⊗Y is dense in C 0 (X,Y ) in the compact-open topology.
As every topological space that has the approximation property has also the strong nonlinear approximation property, we obtain the following corollary. Proof. Exactly as in the proof of Proposition 3.1, we choose any f ∈ C 0 (X,Y ), fix neighbourhood O( f ) of f , and find a compact subset K ⊂ X and a neighbourhood U of zero in Y . Since Y has the approximation property, the map L : Y → Y will be not only continuous and finite-dimensional, but also linear. Therefore, L(θ Y ) = θ Y , which implies that λ i (θ Y ) = 0 for every i ∈ {1, . . . , n}. It is known (see e.g. [8], Part II, Chapter XIII, 4.5) that a continuous linear finite-dimensional map has continuous coordinate functions. Hence, the maps λ i are continuous for every i ∈ {1, . . . , n}. Therefore, we can now proceed as in the proof of Proposition 1 and see that C 0 (X, K) ⊗Y is dense in C 0 (X,Y ) in the compact-open topology.
Next, we shall prove a version of Proposition 3.1 for the case of strongly Klee admissible Hausdorff topological linear spaces. Proof. Note that Y , as a Hausdorff topological linear space, has continuous coordinate functions by Corollary 2.9. Take any f ∈ C 0 (X,Y ) and fix a neighbourhood O( f ) of f in C 0 (X,Y ). Exactly as in the proof of Proposition 3.1, we obtain that there exist a compact subset and L( f (K)) ⊆ Z. Since Y has continuous coordinate functions, the maps λ 1 , . . . , Now, by Lemma 2.11, there exist the extensions λ 1 , . . . , λ n ∈ C(Y, K) of λ 1 , . . . , λ n to the space Y , respectively.
Similarily as in the proof of Proposition 3.1, we define g i = λ i • f ∈ C 0 (X, K) for i ∈ {1, . . . , n} and obtain that Hence, C 0 (X, K) ⊗ Y is dense in C 0 (X,Y ) in the compact-open topology also when Y is a strongly Klee admissible Hausdorff topological linear space.
Recall that for a topological space X, one writes dim(X) = n if n is the smallest nonnegative integer such that for any finite open cover of X one can choose a finite open refinement of that cover such that every x ∈ X is contained in maximally n + 1 elements of that refinement. If there exists a nonnegative integer n such that dim(X) = n, then it is said that dim(X) (or, the topological dimension of X, or the Lebesgue covering dimension of X) is finite. Let us recall that for a map f : X → K from a topological space X to the field K of real or complex numbers, the closure of the set of elements x ∈ X for which f (x) ̸ = 0, was called the support of f and was denoted by supp( f ). In order to prove the next result, we will use another known result. Proof. See the proof of Theorem on slide 10 of [11]. The cited proof copies actually Rudin's ideas of the proof of Theorem 2.13 from [9], p. 40. One has just to notice in the proof of Rudin that the supports supp(h i ) of the constructed functions h i are compact sets (a closed subset of a compact set is compact).
The collection {h 1 , . . . , h n } of functions h i , given in Lemma 3.5, is also called a partition of unity of X. Now we are ready to present a result similar to Theorem 1 (γ) from [2], p. 27.

Proposition 3.6. Let X be a locally compact Hausdorff space and Y a topological linear space. If dim(X)
is finite, then C 0 (X, K) ⊗Y is dense in C 0 (X,Y ) in the compact-open topology.
Proof. As in the proof of Proposition 3.1, fix any f ∈ C 0 (X,Y ), its neighbourhood O( f ), compact subset K ⊂ X, and a neighbourhood U of zero in Y such that f + S(K,U) ⊆ O( f ).
If dim(X) is finite, then there exists a nonnegative integer n such that dim(X) = n. Since the addition is continuous in Y , there exists an open balanced neighbourhood V of zero in Y such that Since dim(X) = n, we can find a finite open subcover D = {O 1 , . . . , O m } of C, which is still a cover of X and where every x ∈ X is contained in maximally n + 1 elements of the cover D. For every x ∈ K, let Then it is clear that the sets I x can have at most n + 1 elements.
For every i ∈ {1, . . . , m}, either there exist k ∈ {1, . . . , l} such that In the first case, take x i = z j , where j ∈ {1, . . . , l} is a minimal such index that O i ⊆ O(z j ). In the second case, take Hence, On the other hand, it is clear that Hence, C 0 (X, K) ⊗Y is dense in C 0 (X,Y ) in the compact-open topology.

APPLICATIONS OF THE DENSITY RESULTS FOR THE CASE OF SEGAL ALGEBRAS
A topological algebra is a topological vector space over K, where the multiplication is separately continuous.

Proposition 4.3. Let X be a locally compact Hausdorff space and A, B topological algebras such that A is a subalgebra of B. If the multiplication in B is jointly continuous, A is a left (right or two-sided) topological
Segal algebra in B via the identity map 1 A , and one of the conditions (a) B has the approximation property, (b) B has the strong nonlinear approximation property and either B is a Hausdorff topological linear space or has continuous coordinate functions, (c) B is a strongly Klee admissible Hausdorff topological algebra, (d) dim(X) is finite is satisfied, then C 0 (X, A) is a left (respectively, right or two-sided) topological Segal algebra in C 0 (X, B) via the identity map 1 C 0 (X,A) .
Proof. As in the proof of Proposition 4.2, we see that in all cases C 0 (X, K) ⊗ B is dense in C 0 (X, B) in the compact-open topology. Hence, the result follows from Corollary 1 in [1]. (2) C 0 (X, A) is a left (respectively, right or two-sided) topological Segal algebra in C 0 (X, B) via 1 C 0 (X,A) .
Proof. As in the proof of Proposition 4.2, we see that in all cases C 0 (X, K) ⊗ B is dense in C 0 (X, B) in the compact-open topology. Hence, the result follows from Corollary 2 in [1].

CONCLUSIONS
We found some sufficient conditions for a Hausdorff space X and a topological linear space Y ensuring that C 0 (X, K) ⊗ Y is dense in C 0 (X,Y ) in the compact-open topology. This allowed us to specify the class of topological algebras for which A is a topological Segal algebra in B if and only if C 0 (X, A) is a topological Segal algebra in C 0 (X, B).