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

1. 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.sub.0](X,Y) the subset of all such f [omega] 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, [[tau].sub.X]), where [[tau].sub.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.sub.0](X,K)[cross product]B (the full definition of this set will be given further in this paper) had to be dense in [C.sub.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)[cross product]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.sub.0](X,K) [cross product]Y is dense in [C.sub.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.

2. PRELIMINARY DEFINITIONS AND RESULTS

Let Y be a topological linear space and K a subset of Y.

Definition 2.1. A map L: K [right arrow]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) [[subset].bar] Z. Moreover, a finite-dimensional mapL:K[right arrow]Y, which can be represented in aform L(y) = [[lambda].sub.1](y)[e.sub.1] + *** + [[lambda].sub.n](y)[e.sub.n] for every y [omega] K, where {[e.sub.1],... ,[e.sub.n]} is a basis of Z, is said to have continuous coordinate functions if the maps [[lambda].sub.i] :K[right arrow] K are continuous for every i [omega] {1,... ,n}. A topological space Y is said to have continuous coordinate functions if every continuous finite-dimensional mapL:Y[right arrow]Y has continuous coordinate functions.

Definition 2.2. It is said that a topological linear space Y is Klee admissible if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional map L:K [right arrow]Y such that L(y)-y[omega] O for every y[omega]K.

Let us recall some definitions of approximation properties.

Definition 2.3. A topological linear space Y has

(a) the approximation property if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional linear map L:Y [right arrow] Y such that L(y) - y [omega] O for every y [omega] K;

(b) the nonlinear approximation property if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional map L:Y[right arrow]Y such that L(y) -y [omega] O for every y [omega] K.

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.

Definition 2.5. We will say that a topological linear space Y

(a) is strongly Klee admissible if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional map L: K[union] {[[theta].sub.Y]} [right arrow]Y such that L([[theta].sub.Y]) = [[theta].sub.Y] and L(y) - y [member of] O for every y [member of] K;

(b) has the strong nonlinear approximation property if for every compact set K [[subset].bar] Y and for every neighbourhood O of zero in Y there exists a continuous finite-dimensional mapL:Y [right arrow] Y such that L([[theta].sub.Y]) = [[theta].sub.Y] and L(y) -y [omega] O for every y [member of] K.

Note that the condition L([[theta].sub.Y]) = [[theta].sub.Y] gives for a finite-dimensional map L : Y [right arrow] Y, which can be written as L(y)=[[lambda].sub.1](y)[e.sub.1] + ***+[[lambda].sub.n](y)[e.sub.n], that [[lambda].sub.1] ([[theta].sub.Y]) = ***=[[lambda].sub.n](y) = [[theta].sub.Y].

Definition 2.6. Let X, Y be topological spaces. It is said that a map f: X [right arrow] Y vanishes at infinity if for every neighbourhood U of zero in Y there exists a compact set K [subset] X such that f(x) [omega] U for every x[omega]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. Iff [member of] [C.sub.0] (X, Y) and [lambda] [member of] C(Y, K) is such that[lambda]([[theta].sub.Y]) = 0, then [lambda] [omicron] f [member of] [C.sub.0](X,K).

Proof Itisclearthat[lambda] f [omicron] C (X,K). Take any neighbourhood O of zero in K. As [lambda]([[theta].sub.Y]) = 0 and [lambda] is continuous, there exists a neighbourhood U of zero in Y such that [lambda](U) [[subset].bar] O. Since f [member of] [C.sub.0](X,Y), there exists a compact set K [[subset].bar] X such that f(x) [member of] U for every x [meber of] X\K. Hence, there exists a compact set K [subset] X such that ([lambda] [omicron] f)(x) = [lambda] (f(x)) [member of] [lambda](U) [[subset].bar] O for every x [member of] X\K . Therefore, [lambda] [omicron] f [member of] [C.sub.0](X,K) . []

Now let us discuss the property of continuous coordinate functions.

Lemma 2.8. Let Y be a Hausdorfftopological linear space, n a positive integer, andL:Y [right arrow] Ya continuous n-dimensional map. Hence, there exists an n-dimensional subspace Z of Y (equipped with the subspace topology) with basis {[e.sub.1]... ,[e.sub.n]}, satisfying L(Y) [[subset].bar] Z such that L(y) = [[lambda].sub.1] (y)[e.sub.1] + ***+ [[lambda].sub.n](y)[e.sub.n] for every y [member of] Y. Then the maps [[lambda].sub.i]:Y [right arrow] K, where i [member of] {1,... ,n}, are continuous, i.e. L has continuous coordinate functions.

Proof. By Theorem 1 from [6], p. 141, we know that Z is isomorphic to [K.sup.n] via the homeomorphism F:Z [right arrow] [K.sup.n], where

[mathematical expression not reproducible]

for all y [member of] Y. Note that the maps [p.sub.i]:[K.sup.n] [right arrow] K, defined by [p.sub.i](([[lambda].sub.1],... , [[lambda].sub.n])) = [lambda]i for every i [member of] {1,... ,n}, are projections, hence, continuous maps.

Using the notations defined above, we see that [[lambda].sub.i]=[p.sub.i] [omicron] F [omicron] L is a continuous map for every i [member of] {1,... ,n} because it is a composition of three continuous maps. Hence, L has continuous coordinate functions. []

Corollary 2.9. Every Hausdorff topological linear space has continuous coordinate functions.

Proof. Let Y be a Hausdorff topological linear space and L : Y [right arrow] Y an arbitrary continuous finite-dimensional map. Then there exist a positive integer n, an n-dimensional subspace Z of Y, and basis {[e.sub.1],... ,[e.sub.n]} of Z such that L(y) = [[lambda].sub.1](y)[e.sub.1] + *** +[[lambda].sub.n](y)[e.sub.n] for every y [member of] Y. But then, by Lemma 2.8, L has continuous coordinate functions. Since L : Y [right arrow] Y was an arbitrary continuous finite-dimensional map, all continuous finite-dimensional maps L : Y [right arrow] 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 [member of] X\Z there is a continuous map f : X [right arrow] [0,1] such that f(x) = 0 and f(z) = 1 for every z [member of] Z.

Lemma 2.11. Let Y be a topological linear Hausdorff space, K a compact subset ofY, and f [member of] C(K,K). Then there exists f [member of] C(Y, K) such that f|[.sub.K]= f, i.e. 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 f [member of] C(K,C). Then we can write f = [f.sub.r] + if, where [f.sub.r],[f.sub.i] [member of] C(K,R) are defined as [f.sub.r](y) = a, [f.sub.i](y) =b for every y [member of] K with f(y) = a + bi. Now, by the first part of the proof, there exist continuous extensions [mathematical expression not reproducible] C(Y, R) of [f.sub.r] and [f.sub.i], respectively. Defining f = [f.sub.r] + [mathematical expression not reproducible], we see that f [member of] C(Y, K) is a continuous extension of f to the whole space Y. []

3. RESULTS CONNECTED WITH THE DENSITY PROPERTY

Let X be a locally compact Hausdorff space and (Y,[[tau].sub.Y]) a topological linear space. Consider the algebra ([C.sub.0](X,Y),[c.sub.Y]) of all continuous maps f : X [right arrow] Y vanishing at infinity equippped with the compact-open topology [c.sub.Y], where the subbase of the topology [c.sub.Y] on [C.sub.0](X, Y) consists of all sets of the form

{S(K, U):K [subset] X, K is compact, U [member of] [[tau].sub.Y]},

where S(K, U) = {f [member of] [C.sub.0](X, Y) : f(K) [[subset].bar] U}.

Defineamap [cross product]:[C.sub.0](X,K) x Y [right arrow] [C.sub.0](X,Y) by

[mathematical expression not reproducible]

for every [phi] [member of] [C.sub.0](X,K),y [member of] Y, and x [member of] X. Let [C.sub.0](X,K) [cross product] Y be the linear span of the set {[phi] [cross product] y : [phi] [member of] [C.sub.0] (X,K), y [member of] Y}.

Next, we will give three results similar to Theorem 1 ([beta]) from [2], p. 27.

Proposition 3.1. Let X be a locally compact Hausdorff space and Y a topological linear space that has the strong nonlinear approximation property. If Y is a Hausdorff space or has continuous coordinate functions, then [C.sub.0](X,K)[cross product]Y is dense in [C.sub.0](X, Y) in the compact-open topology.

Proof. Take any f [member of] [C.sub.0](X, Y) and fix a neighbourhood O(f) of f in [C.sub.0](X, Y). Then there exist a compact subset K [subset] X and a neighbourhood U of zero in Y such that f + S(K, U) [[subset],bar] O(f). Now, f(K) is a compact subset of Y. Since Y has the strong nonlinear approximation property, there exists a continuous finite-dimensional map L : Y [right arrow] Y such that L([[theta].sub.Y]) = [[theta].sub.Y] and L(y) -y [member of] U for every y [member of] f(K). Hence, there exists a positive integer n and a subspace Z of Y with the basis {[e.sub.1],... ,[e.sub.n]} such that L(Y) [[subset].bar] Z.

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) =[[lambda].sub.1](y)[e.sub.1] + *** + [[lambda].sub.n](y)[e.sub.n] and the maps [l[ambda].sub.i] : Y [right arrow] K are continuous with [[lambda].sub.i] ([theta]Y) = 0 for all i [member of] {1,... ,n}.

Take [g.sub.1]=[[lambda].sub.1] [omicron] f,... ,[g.sub.n] = [[lambda].sub.n] [omicron] f. Then, by Lemma 2.7, [g.sub.1],... ,[g.sub.n] [member of] [C.sub.0](X,K).

Note that

[mathematical expression not reproducible]

for each x [member of] K. Hence, for every f [member of] [C.sub.0](X,Y) and every neighbourhood O(f) of f in [C.sub.0](X, Y) there exist integer n >0,[g.sub.1],... ,[g.sub.n] [member of] [C.sub.0](X,K), and [e.sub.1],... , [e.sub.n] [member of] Y such that

[mathematical expression not reproducible]

Therefore, [C.sub.0](X, K) [cross product] Y is dense in [C.sub.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.

Corollary 3.2. Let X be a locally compact Hausdorff space and Y a topological linear space that has the approximation property. Then [C.sub.0](X,K)[cross product]Y is dense in [C.sub.0](X,Y) in the compact-open topology.

Proof. Exactly as in the proof of Proposition 3.1, we choose any f [member of] [C.sub.0](X,Y), fix neighbourhood O(f) of f, and find a compact subset K[subset] X and a neighbourhood U of zero in Y. Since Y has the approximation property, the map L : Y [right arrow] Y will be not only continuous and finite-dimensional, but also linear. Therefore, L([[theta].sub.Y]) = [[theta].sub.Y], which implies that [[lambda].sub.i] ([[theta].sub.Y]) = 0 for every i [member of] {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 [[lambda].sub.i] are continuous for every i [member of] {1,... , n}. Therefore, we can now proceed as in the proof of Proposition 1 and see that [C.sub.0](X, K) [cross product] Y is dense in [C.sub.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.

Proposition 3.3. Let X be a locally compact Hausdorff space and Y a strongly Klee admissible Hausdorff topological linear space. Then [C.sub.0](X,K)[cross product]Y is dense in [C.sub.0](X, Y) in the compact-open topology.

Proof. Note that Y, as a Hausdorff topological linear space, has continuous coordinate functions by Corollary 2.9. Take any f [member of] [C.sub.0](X,Y) and fix a neighbourhood O(f) of f in [C.sub.0](X,Y). Exactly as in the proof of Proposition 3.1, we obtain that there exist a compact subset f(K) [subset] Y, a neighbourhood U of zero in Y, a continuous finite-dimensional map L : f(K) [union]{[theta]Y} [right arrow] Y with L([[theta].sub.Y]) = [[theta].sub.Y], a positive integer n, and a subspace Z of Y with the basis {[e.sub.1],... ,[e.sub.n]} such that f + S(K,U) [[subset].bar] O(f), L(y)-y [member of] U for every y [member of] f (K) and L(f(K)) [[subset].bar] Z. Since Y has continuous coordinate functions, the maps [[lambda].sub.1],... , [[lambda].sub.n] [member of] C(f(K) [union] {[[theta].sub.Y]},K). As L ([[theta].sub.Y]) = [[theta].sub.Y],then [[lambda].sub.1]([[theta].sub.Y]) = *** = [[lambda].sub.n]([[theta].sub.Y])=0.

Now, by Lemma 2.11, there exist the extensions [mathematical expression not reproducible] to the space Y, respectively.

Similarily as in the proof of Proposition 3.1, we define [mathematical expression not reproducible] and obtain that

[mathematical expression not reproducible]

Hence, [C.sub.0](X,K) [cross product] Y is dense in [C.sub.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 [member of] 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.

Remark 3.4. In the mathematical literature, there are actually several definitions of the Lebesgue covering dimension: in some books it is assumed that one can choose an open refinement for any open cover of X, in other books it is assumed that an open refinement should exist only for any finite open cover of X, and in others that a finite open refinement should exist for any finite open cover of X. In Remark 1 in [3], p. 165, it is claimed that the last two definitions coincide.

Let us recall that for a map f : X [right arrow] K from a topological space X to the field K of real or complex numbers, the closure of the set of elements x [member of] X for which f(x) [not equal to] 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.

Lemma 3.5. Let X be a locally compact Hausdorff space, K a compact subset of X, and [V.sub.1],... ,[V.sub.n] open subsets of X such that K [[subset].bar] [V.sub.1] [sunion] [V.sub.2] [union] *** [V.sub.n]. Then there are continuous functions [h.sub.i] :X [right arrow] [0,1], i = 1,2,... ,n, all of compact supports, such that supp([h.sub.i]) [[subset].bar] [V.sub.i], for all i, and

[mathematical expression not reproducible]

for all k [member of] K.

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.sub.i]) of the constructed functions [h.sub.i] are compact sets (a closed subset of a compact set is compact). []

The collection {[h.sub.1],... ,[h.sub.n]} of functions [h.sub.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 ([gamma]) 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.sub.0](X,K)[cross product]Y is dense in [C.sub.0](X,Y) in the compact-open topology.

Proof. As in the proof of Proposition 3.1, fix any f [C.sub.0](X,Y), its neighbourhood O (f), compact subset K [subset] X, and a neighbourhood U of zero in Y such that [omega] + S(K, U) [[subset].bar] 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

[mathematical expression not reproducible]

Now, f(x) + V is an open neighbourhood of f(x) for every x [omega] X. Since f is continuous, then

O(x)=[f.sup.-1](f(x)+V) = {z[omega]X:f(z)[omega]f(x)+V}

is an open neighbourhood of x for every x [omega] X. Hence, the set A = {O(x) : x [omega] K} is an open cover ofK.

As K is a compact set, there exists a finite subcover B = {O([z.sub.1]),... , O([z.sub.l])} of A, with l < [infinity] a positive integer, which is still a cover of K. In a Hausdorff space, every compact set is closed. Hence, X\Kis open and C = {X\K, O([z.sub.1]),... , O([z.sub.l])} is a finite open cover of X.

Since dim(X) = n, we can find a finite open subcover D = {[O.sub.1],... ,[O.sub.m]}of C, which is still a cover of X and where every x [omega] X is contained in maximally n + 1 elements of the cover D. For every x [omega] K, let [I.sub.x] = {i {1,. ..,m}:x [member of] [O.sub.i]}.Thenitis clear that the sets [I.sub.x] can have at most n+1 elements.

As X is a locally compact Hausdorff space and K [subset] X = [O.sub.1] [union] * * * [union] [O.sub.m], then, by Lemma 3.5, there exist a partition of unity [[alpha].sub.1],... ,[[alpha].sub.m] [member of] C(X,[0,1]) [subset] C(X,K) and compact sets [K.sub.1],... ,[K.sub.m](supports of [[alpha].sub.1],... ,[[alpha].sub.m]) such that for every i [omega] {1,... ,m} and every x [member of] X\[K.sub.i] hold [K.sub.i] [[subset].bar] [O.sub.i], [[alpha].sub.i](x) = 0 and for every x [member of]K holds [[alpha].sub.1](x) + *** + [[alpha].sub.m](x) = 1. Hence, [[alpha].sub.1],... ,[[alpha].sub.m] [member of][C.sub.0](X,[0,1]) [subset] [C.sub.0] (X,K).

For every i [omega] {1,... , m}, either there exist k [omega]{ 1,... ,l} such that [O.sub.i] [[subset].bar] O([z.sub.k]) or [O.sub.i] [[subset].bar] X \ K. In the first case, take [x.sub.i] = [z.sub.h] where j [omega] {1,... ,l} is a minimal such index that [O.sub.i] [[subset].bar] O([z.sub.j]). In the second case, take [x.sub.i] = [[theta].sub.X].

Now, for every x [member of] K we have

[mathematical expression not reproducible]

Hence,

[mathematical expression not reproducible]

On the other hand, it is clear that

[mathematical expression not reproducible]

Hence, [C.sub.0](X, K) [cross product] Y is dense in [C.sub.0](X, Y) in the compact-open topology. []

4. 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.

Definition 4.1. A topological algebra (A,[[tau].sub.A]) is a left (right or two-sided) topological Segal algebra if there exists a topological algebra (B, [[tau].sub.B]) and an algebra homomorphism f:A [right arrow] B such that

(1) the image of A by f is dense in B, i.e. [cl.sub.B](f(A)) = B;

(2) [mathematical expression not reproducible]

(3) f(A) is a left (respectively, right or two-sided) ideal of B.

Since for a topological algebra (A, [[tau].sub.A]) there might exist several different topological algebras ( B, [[tau].sub.B]) and algebra homomorphisms f : A [right arrow] B fulfilling the conditions of the definition, we say that "(A, [[tau].sub.A]) is a left (right or two-sided) topological Segal algebra in (B, [[tau].sub.B]) via f : A [right arrow] B", when we want to specify which of the possibly many algebras ( B, [[tau].sub.B]) and mapsf:A [right arrow] B we consider in the particular case.

By Propositions 3.1, 3.3, and 3.6, we will have now some new results for topological Segal algebras.

Proposition 4.2. Let X be a locally compact Hausdorff space, A,B topological algebras, and[iota] :A [right arrow] B a map. Consider the algebras ([C.sub.0](X,A),[c.sub.A]) and ([C.sub.0](X,B),[c.sub.B]) equipped with the compact-open topologies [c.sub.A] and [c.sub.B] and define a map [omega]: [C.sub.0](X,A) [right arrow] [C.sub.0](X,B) by [omega](f) := [iota] [omicron] f for every f [member of] [C.sub.0](X,A). Suppose that the multiplication in B is jointly continuous, [iota] is continuous and open algebra monomorphism, and [iota] (A) is a dense left (right or two-sided) ideal ofB. If 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.sub.0](X,A) is a left (respectively, right or two-sided) topological Segal algebra in [C.sub.0](X,B) vis [omega].

Proof. Using Corollary 3.2 in case (a), Proposition 3.1 in case (b), Proposition 3.3 in case (c), and Proposition 3.6 in case (d), we see that in all cases [C.sub.0](X,K) [cross product] B is dense in [C.sub.0](X,B) in the compact-open topology. Hence, the result follows from Proposition 3 (f) in [1]. []

Proposition 4.3. Let X be a locally compact Hausdorff space and A, B topological algebras such that A is a subalgebra ofB. 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.sub.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.sub.0](X,A) is a left (respectively, right or two-sided) topological Segal algebra in [C.sub.0](X,B) via the identity map [mathematical expression not reproducible].

Proof. As in the proof of Proposition 4.2, we see that in all cases [C.sub.0](X, K) [cross product] B is dense in [C.sub.0](X,B) in the compact-open topology. Hence, the result follows from Corollary 1 in [1]. []

Proposition 4.4. Let X be a locally compact Hausdorff space, A, B topological algebras such that A is a subalgebra of B and the multiplication in B is jointly continuous. Consider the algebras ([C.sub.0](X,A),[c.sub.A]) and ([C.sub.0](X,B),[c.sub.B]) equippped with the compact-open topologies cA andcB, respectively. Suppose that 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 the following conditions are equivalent:

(1) A is a left (right or two-sided) topological Segal algebra in B via [1.sub.A];

(2) [C.sub.0](X,A) is a left (respectively, right or two-sided) topological Segal algebra in [C.sub.0](X,B) via [mathematical expression not reproducible].

Proof. As in the proof of Proposition 4.2, we see that in all cases [C.sub.0](X,K) [cross product] B is dense in [C.sub.0](X,B) in the compact-open topology. Hence, the result follows from Corollary 2 in [1]. []

5. CONCLUSIONS

We found some sufficient conditions for a Hausdorff space X and a topological linear space Y ensuring that [C.sub.0](X, K) [cross product] Y is dense in [C.sub.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.sub.0](X,A) is a topological Segal algebra in [C.sub.0](X,B).

ACKNOWLEDGEMENTS

The research was supported by the institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The author would like to thank all of the participants of the seminar 'Types of Topological algebras', held in the University of Tartu in spring 2017, for the questions, remarks, and discussions, which helped to improve the present paper and to make it more general. The author also wants to thank the referees who pointed out some gaps in the proofs and suggested reference [10].

REFERENCES

[1.] Abel, Mart. Generalisation of Segal algebras for arbitrary topological algebras. Period. Math. Hung. https://link.springer.com/article/10.1007/s10998-017-0222-z (accessed 2018-05-11).

[2.] Abel, Mati. The denseness everywhere of subsets in some spaces of vector-valued functions (in Russian). Tartu Riikl. Ul. Toimetised, 1987, 770, 26-37.

[3.] Aleksandrov, P. S. and Pasynkov, B. A. Introduction to Dimension Theory (in Russian). Nauka, Moscow, 1973.

[4.] Gillman, L. and Jerison, M. Rings of Continuous Functions. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N.J.--Toronto--London--New York, 1960.

[5.] Gorniewicz, L. and Slosarski, M. Fixed points of mappings in Klee admissible spaces. Control Cybernet., 2007, 36(3), 825-832.

[6.] Horvath, J. Topological Vector Spaces and Distributions. Vol. I. Addison-Wesley Publishing Co., Reading, Mass.--London--Don Mills, Ont., 1966.

[7.] Husain, T. Introduction to Topological Groups. R. E. Krieger Publ. Co., Philadelphia, 1981.

[8.] Kantorovich, L. V. and Akilov, G. P. Functional Analysis. (Translated form the Russian by Howard L. Silcock). Second Edition. Pergamon Press, Oxford--Elmsford, N. Y., 1982.

[9.] Rudin, W. Real and Complex Analysis. McGraw-Hill Book Co., New York--Toronto, Ont.--London, 1966.

[10.] Shuchat, A. H. Approximation of vector-valued continuous functions. Proc. Am. Math. Soc., 1972, 31(1), 97-103.

[11.] Thomsen, K. More on locally compact Haudorff spaces, http://data.math.au.dk/kurser/advanalyse/F06/lecture11pr.pdf (2005) (accessed 2017-02-03).

[12.] Waelbroeck, L. Topological vector spaces. In Summer School on Topological Vector Spaces. Held at the Universite Libre de Bruxelles, Brussels, in September 1972 (Waelbroeck, L., ed.). Lecture Notes in Math., 331. Springer-Verlag, Berlin--New York, 1977, 1-40.

Mart Abel

School of Digital Technologies, Tallinn University, Narva 29, 10120 Tallinn, Estonia;

Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia; mart.abel@tlu.ee, mart.abel@ut.ee

Received 17 November 2017, revised 22 March 2018, accepted 28 March 2018, available online 20 June 2018

(1) In some books (see, for example, [8]) it is assumed that a finite-dimensional map has to be also continuous.

Koikjal tiheduse omadusest pidevate lopmatuses haabuvate funktsioonide algebras

Mart Abel

Olgu K kas reaalarvude voi kompleksarvude korpus ja Y topoloogiline vektorruum ule K. Artiklis on leitud moningad piisavad tingimused topoloogilise ruumi X ja topoloogilise algebra Y jaoks, mille korral algebra [C.sub.0](X,K) [cross product]Y on koikjal tihe pidevate lopmatuses haabuvate funktsioonide algebras [C.sub.0](X,Y).

Nende tulemuste rakendusena saadakse, et topoloogiline algebra A on topoloogiline Segali algebra topoloogilises algebras B siis ja ainult siis, kui pidevate lopmatuses haabuvate funktsioonide algebra [C.sub.0](X,A) on topoloogiline Segali algebra pidevate lopmatuses haabuvate funktsioonide algebras [C.sub.0](X,B).

https://doi.org/10.3176/proc.2018.3.07
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Author:Abel, Mart
Publication:Proceedings of the Estonian Academy of Sciences
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Date:Sep 1, 2018
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