AN ASCOLI THEOREM FOR SEQUENTIAL SPACES

Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompactness,” with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context of C∗-algebras. 2000 Mathematics Subject Classification. 54A20, 54C35, 54E15.

1. Introduction.Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category.Such general theorems are inspired by the classical Ascoli theorem, proved by G. Ascoli (and independently by C. Arzelà) in the 19th century (see [3,4]).It characterizes compactness of sets of continuous real-valued functions on the interval [0, 1] with respect to the topology of uniform convergence.Since then, many related theorems have been proved, for example, characterizing compactness of sets of continuous functions from a topological to a uniform space (see [6]), of uniformly continuous functions from a merotopic to a uniform space (see [5]) of continuous functions between topological spaces (see [14,17]).As was pointed out by Wyler in [24], it is clear that the setting for Ascoli theorems requires natural function space structures; the existence of nice function spaces is guaranteed by Cartesian closedness of the considered topological construct.Around 1980, Dubuc [8] and Gray [12] both proposed a general theory for Ascoli theorems in a categorical setting, but as neither of them seems to be entirely satisfactory, Wyler suggested that more examples should be constructed in order to guide the general theory.Wyler himself developed new examples of Ascoli theorems for sets of continuous functions between limit spaces and of uniformly continuous functions from a uniform convergence space to a pseudo-uniform space (see [24]).
In this paper, we present another setting for an Ascoli theorem: we choose the construct L of sequential spaces.Sequential spaces were already introduced at the beginning of the century by Fréchet (see [9,10]) and Urysohn (see [2,23]), even before topological spaces were axiomatized.Since then they have extensively been used as a tool in topology and analysis.Since the 60s sequential structures have been investigated from a categorical point of view (for categorical background, we refer to [1]); in particular, it was shown that the construct L is Cartesian closed.The natural function spaces available in the setting of L are extensively used in our development of the theory.We define appropriate concepts of equicontinuity and even continuity in L, and study the relations between these concepts.Further, we investigate relations between the different sequential function space structures (pointwise convergence, continuous convergence, and uniform convergence) and look at the induced structures on equicontinuous and evenly continuous sets.In this way, we obtain two versions of an Ascoli theorem for sets of continuous functions from an L-space to a uniform space.Finally, we apply our theorem in an example in the context of Ꮿ * -algebras.
The set N is the set of nonnegative integers and MON s the set of all strictly increasing mappings from N to N. If ξ is a sequence in a set X, we often write ξ n for ξ(n), and the sequence itself is denoted by ξ n .The Fréchet-filter of ξ on X is denoted by Ᏺ(ξ), that is, the filter generated by the sets Then (X, ᏸ) is called an L-space or sequential (convergence) space.As usual, such a space will often be denoted by its underlying set only.
we say that ξ ᏸ-converges to x (or simply X-converges to x or converges to x) and that x is an ᏸ-limit point of ξ.If X and Y are L-spaces, a function f : and continuous if it is continuous at each point of X.The set of all continuous functions from X to Y is denoted by C(X, Y ); it is a subset of F(X,Y ), the set of all functions from the set X to the set Y .The construct of all L-spaces and continuous maps as morphisms is denoted by L. It is a well-fibred topological construct.A source Often, we will work in L * , the bireflective subconstruct of L with as objects all L-spaces (X, ᏸ) satisfying the Urysohn-axiom: (ᏸ is then called an L * -structure on X).Objects of L * are called L * -spaces or Urysohn sequential (convergence) spaces.For example, if (X, ᐁ) is a uniform space, with Ꮾ a base for ᐁ, an L * -structure on X is defined by (it is the convergence of sequences in the topology on X induced by ᐁ, and it is independent of the choice of the base Ꮾ).In the following, if a uniform space (X, ᐁ) is considered as an L * -space, X will always be endowed with the above-mentioned L * -structure.If (X, ᏸ) is an L-space, a pretopological structure P (ᏸ) on X is defined (see [15]) by the closure-operator (1.7) For a subset A of an L-space (X, ᏸ), cl ᏸ A (or simply cl A) always means the closure of A in the pretopological space (X, P (ᏸ)) and is the neighborhood filter of x in (X, P (ᏸ)).A subset D of an L-space (X, ᏸ) is called dense if it is dense in (X, P (ᏸ)), that is, if each point in X is a limit point of a sequence in D. An L-space (X, ᏸ) is called separable if there is a countable dense subset in (X, ᏸ), that is, if (X, P (ᏸ)) is separable.A neighborhood covering system (or shortly ncs) of an L-space (X, ᏸ) is a neighborhood covering system of the pretopological space (X, P (ᏸ)), that is, a set σ of subsets of X that contain a neighborhood of each point of X.
A bornology (see [13]) of a set X is a subset α of ᏼ(X) with (i) A ∈ α, A ⊂ A ⇒ A ∈ α, (ii) finite unions of sets in α are in α, (iii) all finite subsets of X are in α.A set with a bornology is called a bornological set.Bornological sets are objects of a topological construct Born.A morphism f : (X, α) → (X, β) in Born is a mapping f : X → Y with f (α) ⊂ β.

Compactness and precompactness for
We now define the concept of precompactness in L which, according to Wyler (see [24]), should be a functor L → Born preserving the underlying sets and mappings.

Definition 2.2. A subset A of an L-space is called precompact if each sequence in
A has a subsequence that converges to a point of X.
Evidently, compact subsets of an L-space are precompact.It is easily seen that a subset of a precompact subset of an L-space still is precompact, that all finite subsets of an L-space are precompact and that the union of a finite number of precompact subsets again is precompact.Also we have that, if f : X → Y is a continuous function between L-spaces, the image f (A) by f of a precompact subset A of X is precompact in Y .Thus precompact subsets define a functor Pr : L → Born, which preserves underlying sets and mappings.We can use Example 5.3 in [22] to show that Pr does not preserve products.
3. Uniformizable, regular, and R 0 spaces.The following definitions are inspired by the concepts of R 0 -limit spaces in [21] and uniformizable limit spaces in [24].
and it is called uniformizable if it satisfies Uniformizable spaces clearly are R 0 -spaces.The full subcategories of L with as objects all R 0 -spaces (resp., all uniformizable spaces) are bireflective in L.
In [11] a notion of regularity for L-spaces is defined.We formulate this definition here in the context of L * -spaces.Definition 3.2.Let X be an L * -space.Take ξ ∈ X N , x ∈ X, and Ξ n ∈ (X N ) N .We say that Ξ n links ξ and x if for each k ∈ N, the sequence Ξ k is X-converging to ξ k and for each f ∈ N, the sequence Ξ n (f (n)) is X-converging to x; in this case ξ and x are said to be linked.
and only if ξ and x are linked, for all ξ ∈ X N and x ∈ X.

Function spaces in L.
We first introduce some L-structures on function spaces.
(1) If X is a set and Y is an L-space, the L-structure π of pointwise convergence on F(X,Y ) is the product structure in L on F(X,Y ), that is, for a sequence f n in F(X,Y ) and f ∈ F(X,Y ), we have If Y is an L * -space, then so is (F (X, Y ), π ).
(2) If X and Y are L-spaces, the L-structure Γ of continuous convergence on C(X, Y ) is defined by (3) If X is a set and (Y , ᐁ) is a uniform space with Ꮾ a base for ᐁ, and if A ⊂ X, the sets for B ∈ Ꮾ form a base for a uniformity on F(X,Y ), which induces an L * -structure on F(X,Y ).It does not depend on the choice of the base Ꮾ.We call it the sequential structure of uniform convergence on A, and denote it by s X,Y ᐁ,A , or simply s ᐁ,A .For a sequence f n in F(X,Y ) and f ∈ F(X,Y ), we have If A = X, we write s ᐁ instead of s ᐁ,A , and call s ᐁ the sequential structure of uniform convergence.If now σ ⊂ ᏼ(X), we define s X,Y σ (or s σ ) as the supremum in L * of all the structures s ᐁ,A on F(X,Y ), with A ∈ σ ; s σ is called the structure of uniform convergence on the sets of σ .For a sequence f n in F(X,Y ) and f ∈ F(X,Y ), we have it is easily seen that is an initial source in L * .Even more, each map is initial.Finally, remark that, if σ = {A} with A ⊂ X, then s σ = s ᐁ,A , and that, if σ = {{x}; x ∈ X}, then s σ = π .
For the above-defined function convergences π , Γ , and so forth, we use the same symbols for their induced convergences on subsets of their definition sets, that is, if X and Y both carry L-structures, we use π for the subspace structure of the sequential structure of pointwise convergence on the subset C(X, Y ) of F(X,Y ).
We recall that L and L * are Cartesian closed topological constructs and that, for L-spaces X and Y , (C(X, Y ), Γ ) is the corresponding power-object (see [1]): We now investigate limits of sequences of continuous functions in the sequential structures of uniform convergence.Proposition 4.2.Let X be an L-space, (Y , ᐁ) a uniform space, x ∈ X and A ∈ ᐂ(x).
Corollary 4.3.Let X be an L-space and let (Y , ᐁ) be a uniform space.If σ is an ncs of X, then C(X, Y ) is closed in (F (X, Y ), s σ ).
In the following theorem we discuss some relations between the different function space structures on F(X,Y ).
5.Even continuity.In [14], Kelley defined even continuous mappings between topological spaces, and Poppe [19] generalized this notion to sets of continuous mappings between limit spaces.The following is straightforward transcription of these definitions to L-spaces.
It is easily seen that a subset of an evenly continuous set in C(X, Y ) is still evenly continuous and that a finite union of evenly continuous subsets is still evenly continuous.This does not mean that the evenly continuous subsets of C(X, Y ) always form a bornology on C(X, Y ); the next proposition clarifies when they do.
Proposition 5.2.The following statements are equivalent: (a) The L-space Y is an R 0 -space.(b) For all L-spaces X the evenly continuous subsets of C(X, Y ) form a bornology on C(X, Y ).Proposition 5.3.For a fixed R 0 -space Y , even continuity defines a functor EC Y : L op → Born, with EC Y (X) = (C(X, Y ), {evenly continuous subsets}).

Equicontinuity.
In this section, X is an L-space and (Y , ᐁ) is a uniform space.
The subset H is called equicontinuous in A ⊂ X if H is equicontinuous at each point x of A, and H is called equicontinuous if H is equicontinuous in X.
Proposition 6.2.The subsets of C(X, Y ) that are equicontinuous in x form a bornology on C(X, Y ).Proposition 6.3.An equicontinuous subset of C(X, Y ) is evenly continuous.Remark 6.4.To see that an evenly subset of C(X, Y ) need not be equicontinuous, consider the following example, based on an example of Poppe in [20].Let X be the L *space with the underlying set Q ∩ [0, 1] and with the sequential structure induced by the usual metric on Q.Further, let Y be the uniform subspace of R with as underlying set all strictly positive rational numbers.Then take an irrational number i 0 ∈ [0, 1] and a strictly decreasing sequence r n of rationals in [0, 1], converging in R to i 0 .Now for all n ∈ N define a continuous function Finally put H = {f n ; n ∈ N}.
We first show that H is not equicontinuous at 0. If ξ is the sequence in X with ξ n = 1/n for n ∈ N 0 and ξ 0 = 0, then ξ → x.For n ∈ N 0 we now have To prove that H is evenly continuous, take x ∈ X, y ∈ Y , ξ ∈ X N , and a sequence →y clearly implies that the set {r s(n) ; n ∈ N} is finite, and so that also and so the set {f s(n) ; n ∈ N} is finite, otherwise the sequence f s(n) (x) has an infinite number of terms greater than y + 1, which contradicts the convergence

So in each case the set {f s(n)
; n ∈ N} is finite, and so evenly continuous.This means that f s(n) (ξ n ) Y →y.So we have proved that H is evenly continuous.Note that is not precompact in Y .This is not surprising, for the following proposition shows that H would be equicontinuous in 0 if H(0) were to be precompact in Y , since H is evenly continuous.
Proof.Suppose H is not equicontinuous in x.Then there exists U ∈ ᐁ and a sequence ξ with ξ X →x such that yielding a contradiction.
The following proposition reduces equicontinuity of a set of functions to an ordinary continuity of a suitable function, as one can do in the context of topological spaces [14].First we need a few notations and remarks.
For x ∈ X, x is the evaluation map in x x : Proof.First remark that for a sequence ξ in X, we have Now λ H is continuous in x if and only if, for all sequences ξ converging to x in X, we have and with the equivalences above, this is precisely the condition for H to be equicontinuous in x.

Convergence on evenly continuous and equicontinuous sets
Proof.Take x ∈ X, ξ ∈ X N with ξ X →x and let f n ∈ H N be a sequence in H with f n π →f .For all n ∈ N, we put We now prove that Ξ links f • ξ and f (x) in Y .For all n ∈ N, Ξ n is a subsequence of f k (ξ n ) k , and we inductively define t : N → N as follows: Then clearly t also is a strictly increasing function N → N. Now f t(n+1) is a subsequence of f n and f t(n+1) π → f , and is evenly continuous.Now and so , and so f is continuous.Proposition 7.4.Let X be a compact L-space, and (Y , ᐁ) a uniform space.If and (Y , ᐁ) is a uniform space, then the closures F(X,Y ) of an evenly continuous subset of C(X, Y ) in the sequential structures π and s ᐁ coincide.Proposition 7.6.Let X be an L-space, x ∈ X, and let (Y , ᐁ) be a uniform space.If H is a subset of C(X, Y ), then H is equicontinuous in x if and only if cl π H is equicontinuous in x.
8.An extension theorem.In this section, X is an L-space and (Y , ᐁ) is a uniform space.Recall that ξ ∈ Y N is a Cauchy-sequence if and only if ∀U ∈ ᐁ, ∃n 0 ∈ N : p, q ≥ n 0 ⇒ ξ p ,ξ q ∈ U.
(8.1) Proposition 8.1.Let ξ X → x and let f n be a sequence in C(X, Y) with {f n ; n ∈ N} equicontinuous in x.If for all k ∈ N, f n (ξ k ) n is a Cauchy-sequence in (Y , ᐁ), then f n (x) also is a Cauchy-sequence in (Y , ᐁ). ) Proof.This theorem is an immediate consequence of the previous proposition.

An Ascoli theorem
Proposition 9.1.Let X and Y be L-spaces with Y an L * -space, which is This proves because Y is an L * -space.
Theorem 9.2.Let X and Y be L-spaces with Y an L * -space, which is R 0 .If H is precompact in (C(X, Y ), Γ ), then H is evenly continuous and for all x ∈ X, H(x) is precompact in Y .
Proof.The proof follows from Proposition 9.1 and the continuity of and H(x) = x(H).
Theorem 9.3.Let X be an L-space and (Y , ᐁ) a uniform space.If H is precompact in (C(X, Y ), Γ ), then H is equicontinuous and for all x ∈ X, H(x) is precompact in Y .Further, if X is a separable L-space, these two conditions are also sufficient for H to be precompact in (C(X, Y )Γ ).
Proof.The first part of the theorem follows from Theorem 9.2 and Proposition 6.5.For the second part, suppose H is equicontinuous and H(x) is precompact in Y for all x ∈ X, and let D = {d n ; n ∈ N} be a dense subset of X.Take a sequence f n in H.Because f n (d 0 ) is a sequence in H(d 0 ), it has a convergent subsequence f s 0 (n) (d 0 ) (s 0 ∈ MON s ); denote the limit point of this subsequence by f (d 0 ).Now suppose, by induction, that s k ∈ MON s and . This defines a function f : D → Y .Now for all k ∈ N there is a t ∈ MON s such that for all n ≥ k, This means that f sn(n) (d k ) n≥k is a subsequence of f s k (n) (d k ) , and thus The extension theorem (Theorem 8.3) then gives a continuous extension f : Corollary 9.4.Let X be a compact and separable L-space and (Y , ᐁ) a uniform space.Then a subset Proof.The proof follows from Theorem 9.3 and Theorem 4.4(g).
We can easily rewrite Theorem 9.3 in another form.Therefore, if F is a set, X and Y are L-spaces, a map Φ : For such a dual map, we put Γ Φ , the initial L-structure, on F for the map F → C(X, Y ) : f → Φ f .If Y is an L * -space, Γ * -space, Γ Φ is an L * -structure on F .Finally, for H ⊂ F , we set Φ H = {Φ f ; f ∈ H}.Theorem 9.5.Let F be a set, X an L-space, (Y , ᐁ) a uniform space, and Φ : X ×F → Y a dual map.If H is precompact in (F , Γ Φ ), then Φ H is equicontinuous in C(X, Y ) and for all x ∈ X, Φ H (x) is precompact in Y .Further, if X is separable and cl Γ Φ Φ H ⊂ Φ F , then these two conditions are also sufficient for H to be precompact in (F , Γ Φ ).
10.An example.We give an example of an application of our Ascoli theorem in the context of Ꮿ * -algebras.Some experience with the fundamental parts of the theory of Ꮿ * -algebras is needed and can be found in [7,18].The example is based on an application of another Ascoli theorem for "pseudotopological classes" by McKennon in [16].
Let A be a Ꮿ * -algebra, which we may and will regard as a subalgebra of its enveloping von Neumann algebra A ν .Let Q A = {x ∈ A ν ; ∀a, b ∈ A : a * xb ∈ A} be the set of quasi-multipliers of A and S A the set of all states of A (i.e., the set of all positive linear functionals on A with norm 1).Each linear functional ϕ : A → C has a unique linear extension ϕ ν : A ν → C that is continuous for the weak topology on A ν , and for x ∈ Q A the map x : A → C : ϕ → ϕ ν (x) belongs to the bidual of A. On S A we place the L * -structure introduced by the weak * -topology on A relativized to S A .In this case, we can show that Φ : S A × Q A → C is a dual map.Furthermore, norm-bounded sequences in Q A converge in Γ Φ if and only if they converge in the quasi-strict topology on Q A , that is, the topology on Q A induced by all semi-norms Q A → C : x → a * xa for a ∈ A. If the C * -algebra A is separable, then S A is also separable.And finally, for a norm-bounded subset H of Q A , the condition cl Γ Φ Φ H ⊂ Φ F is satisfied.So we can apply Theorem 9.5 and get the following theorem.
Theorem 10.1.If A is a separable Ꮿ * -algebra, then for each norm-bounded subset H of Q A the following are equivalent: (1) H is precompact in the sequential structure induced by the quasi-strict topology on Q A .
(2) { x| S A ; x ∈ H} is equicontinuous in C(S A , C) and for all ϕ ∈ S A the set {ϕ ν (x); x ∈ H} is precompact in C.

. 1 ) 7 . 2 .
Corollary If X and Y are L-spaces, then the closures in C(X, Y ) of an evenly continuous subset of C(X, Y ) for the sequential structures π and Γ coincide.Proposition 7.3.Let X and Y be L-spaces with Y a regular L .10)If H is endowed with the structure π of pointwise convergence, it is easily seen that all functions x| H : H → Y are continuous.So we have a function λ H : X → C(H, Y ) : x → x| H . (6.11) Proposition 6.6.If H ⊂ C(X, Y ) and x ∈ X, the following are equivalent: Let D be a dense subset of X, f n a sequence in C(X, Y ) with