- Open Access
Ascoli-type theorems in the cone metric space setting
Journal of Inequalities and Applications volume 2014, Article number: 420 (2014)
We give some necessary and sufficient conditions for (global) continuity of the limit of a pointwise convergent net of cone metric space-valued functions, defined on a Hausdorff topological space, in terms of weak filter exhaustiveness. In this framework, we prove some Ascoli-type theorems, considering also possibly asymmetric and extended real-valued distance functions. Furthermore, we pose some open problems.
MSC:26E50, 28A12, 28A33, 28B10, 28B15, 40A35, 46G10, 54A20, 54A40, 06F15, 06F20, 06F30, 22A10, 28A05, 40G15, 46G12, 54H11, 54H12, 47H10.
In the literature there have been several studies about cone metric spaces, namely abstract structures endowed with a distance function taking values in an ordered vector or a normed space, which includes in particular metric semigroups, whose an example is the set of fuzzy numbers, which is not a group (see for instance [1–5]). These structures are closely related with order vector spaces endowed with abstract convergences satisfying suitable axioms, but in which in general convergence of subsequences of convergent sequences is not required, like filter convergence (see also [6–12]). A comprehensive historical survey on main properties of these structures and several (recent) results about abstract convergences, distances with values in normed, solid or Hausdorff topological vector spaces and fixed point theorems, which have several applications to differential, functional and stochastic equations and reconstruction of signals can be found in [13–29]. In this paper we investigate some properties of continuity of the limit of a net of functions, taking values in cone metric spaces, in terms of weak filter exhaustiveness, extending earlier results proved in [30–32], and relate filter exhaustiveness with filter uniform convergence (on compact subsets). Moreover, we give some Ascoli-type theorems for lattice group-valued functions defined on metric or topological spaces, extending previous results proved in  (see also ), and consider also asymmetric distances (see also [32, 34]) and extended real-valued distances, like Lipschitz metrics, dealing with functions which are not necessarily contractions and extending earlier results proved for real-valued or metric space-valued functions in [30, 32] and , respectively. We present some examples to support the results obtained in our setting. Asymmetric distance has different applications in several branches of mathematics and in physics, for example in gradient flow models (see also  and the bibliography therein), and is related also with the study of several semicontinuity properties of functions (see for instance ). Observe that extended Lipschitz metrics are complete and extended Lipschitz metric convergence is, in general, strictly stronger than uniform convergence on bounded sets. Moreover, extended Lipschitz metrics are equivalent with the supremum metrics when the topological space X in which the involved functions are defined is bounded and uniformly discrete (in its own metric), and they are also equivalent with the Sherbert and Weaver metrics when X is just bounded (see also [35, 37–39]). Furthermore, note that, since in lattice groups the order convergence is in general not generated by any topology (see also ), in our context it is not advisable to deal with concepts like closedness and compactness in terms of topologies. So we formulate the corresponding notions directly in the setting of convergence and in terms of function nets, including the classical concepts as particular cases and giving some relations between filter pointwise convergence and filter uniform convergence on compact sets. In the literature, there have been several recent studies about abstract Ascoli-type theorems, which extend earlier results given in . In , different Ascoli-type theorems are proved, in connection with various kinds of convergence and exhaustiveness of function nets. In  and  these convergences, together with the concept of exhaustiveness, are considered in the filter/ideal context, and in this setting some Ascoli-type theorems for real-valued functions are extended. Some other versions of the Ascoli theorem can be found, for instance, in [34, 44–46]. Our approach is direct, simple and easy to handle in the context of our considered structures, that is, when it is dealt with nets of functions taking values in cone metric spaces, defined in general Hausdorff topological spaces and with filter exhaustiveness instead of metric spaces and equicontinuity, respectively, and it allows us to give direct necessary and sufficient conditions. We consider symmetric or asymmetric distances with values in lattice groups and use the tool of (weak) filter exhaustiveness in connection with (global) continuity of the limit function and uniform convergence on compact sets. One of the main used methods is to use some kinds of convergence of suitable subnets of the given net to deduce some compactness properties. This is given in a very abstract context, comparing two kinds of compactness for function nets, and after a particular case is presented, using compactness of suitable sets, properties of convergence and boundedness in metric spaces and the Tychonoff theorem. Furthermore we consider Lipschitz-type metrics using completeness properties an a ‘total boundedness’ argument in terms of subsequence, without using a topological approach. Finally, we pose some open problems.
We begin with some fundamental properties of convergence and continuity in the lattice group context.
A nonempty set is said to be directed iff ≥ is a reflexive and transitive binary relation on Λ, such that for any two elements there is with and .
A cone metric space is a nonempty set R endowed with a function , where Y is a Dedekind complete lattice group, satisfying the following axioms:
and if and only if ;
(triangular property), for all ,
(see also [11, 12]). Such a function ρ will be called a distance function. If ρ satisfies only the first and the third of the above axioms, but not necessarily the symmetric property, then we say that ρ is an asymmetric distance function and that is an asymmetric cone metric space (for literature on the asymmetric case, see also  and the related bibliography). Note that any Dedekind complete -group Y is a cone metric space: indeed, it is enough to take , (the absolute value).
Example 2.1 Let T any arbitrary nonempty set, , and fix a positive real number . For every and , put
and set . It is not difficult to check that d is an asymmetric distance function (see also ).
Let R be a (possibly asymmetric) cone metric space and Y be its associated Dedekind complete -group. A sequence in Y is called an -sequence iff it is decreasing and . A net in R (that is an indexed system of elements of R such that the index set Λ is directed) is forward order convergent or forward -convergent (resp. backward order convergent or backward -convergent) to iff there exists an -sequence in Y such that for every there is with (resp. ) for all , . We say that order converges or -converges to iff it is both forward and backward -convergent to x, and in this case we will write .
Let X be a Hausdorff topological space. A function is said to be forward (resp. backward) continuous at a point iff there exists an -sequence in Y (depending on x) such that for every there is a neighborhood of x with (resp. ) whenever .
A function is globally forward (resp. backward) continuous on X iff there is an -sequence in Y such that for any and there is a neighborhood of x with (resp. ) for each . We say that is (globally) continuous on X iff it is both (globally) forward and (globally) backward continuous on X.
We now recall some basic notions on ideals and filters.
Let Λ be any nonempty set, and be the class of all subsets of Λ. A family of sets is called an ideal of Λ iff whenever and for each and we get . A class of sets is a filter of Λ iff for all and for every and we have .
An ideal ℐ (resp. a filter ℱ) of Λ is said to be non-trivial iff and (resp. and ).
Let be a directed set. A non-trivial ideal ℐ of Λ is said to be -admissible iff for each , where .
A non-trivial filter ℱ of Λ is -free iff for every .
Given an ideal ℐ of Λ, we call dual filter of ℐ the family . In this case we say that ℐ is the dual ideal of ℱ and we get .
When endowed with the usual order, the -admissible ideals and the -free filters are called simply admissible ideals and free filters, respectively. The filter is the filter of all subsets of ℕ whose complement is finite, and its dual ideal is the family of all finite subsets of ℕ. The filter is the filter of all subsets of ℕ having asymptotic density 1, while its dual ideal is the family of all subsets of ℕ, having null asymptotic density. Note that is a P-filter, namely a filter ℱ of ℕ such that for every sequence in ℱ there is another sequence in ℱ, such that the symmetric difference is finite for all and (see also [33, 47]).
A nonempty family is said to be a filter base of Λ iff for every there is an element with . Note that, if is a filter base of Λ, then the family is a filter of Λ. We call it the filter generated by .
If , then is a filter base of Λ, and the filter generated by is a -free filter of Λ (see also ).
Let ℱ be a -free filter of Λ and choose . A net in R is said to be ℱ-forward bounded (resp. ℱ-backward bounded) with respect to iff there is , such that (resp. ). We say that is ℱ-bounded with respect to iff it is both ℱ-forward and ℱ-backward bounded with respect to , and that is bounded (resp. forward bounded, backward bounded) iff it is -bounded (resp. -forward bounded, -backward bounded).
We now give the fundamental notions of filter convergence and related topics in the cone metric space setting. Without loss of generality, we consider the symmetric case. Analogously it is possible to deal with the corresponding ‘backward’ and ‘forward’ concepts in the asymmetric case (see also ).
A net in a cone metric space R -converges to (shortly, ) iff there exists an -sequence in Y with for each . A net in R is -Cauchy iff there is an -sequence in Y such that for every there is with for each . Note that, since R is Dedekind complete, a net in R is -convergent if and only if it is -Cauchy (see also [33, 48]).
Let Ξ be any nonempty set. We say that a family in R -converges to uniformly with respect to (shortly, -converges to ) as λ varies in Λ iff there is an -sequence in Y with
A family -converges to (as λ varies in Λ) iff there exists an -sequence in Y such that for each and we get . By -convergence we will denote the -convergence. Observe that, when , -convergence coincides with usual filter convergence (see also [8, 9, 33]).
Let . A net , , is said to be ℱ-exhaustive at x iff there is an -sequence such that for any there exist a neighborhood U of x and a set such that for each and we have .
A net , , is weakly ℱ-exhaustive at x iff there is an -sequence such that for each there is a neighborhood U of x such that for every there is with whenever .
We say that , , is (weakly) ℱ-exhaustive on X iff it is (weakly) ℱ-exhaustive at every with respect to a single -sequence, independent of .
Similarly as above it is possible to formulate the concepts of (weak) ℱ-forward (backward) exhaustiveness (see also ).
Of course the concepts of (weak, forward, backward) filter exhaustiveness can be given also analogously for sequences of functions, by taking with the usual order.
In the next section we will see that, in general, the notion of weak ℱ-exhaustiveness is strictly weaker than that of ℱ-exhaustiveness (for the case see also [, Remark 2.8]).
3 The main results
We now give, in the context of filter convergence and lattice groups, a necessary and sufficient condition under which the limit of a pointwise convergent net is (globally) continuous, extending [, Proposition 17] to nets of lattice group-valued functions.
Theorem 3.1 Under the same above notations and assumptions, let ℱ be a -free filter of Λ, fix , and suppose that , , -converges to on X with respect to a single -sequence in Y. Then the following are equivalent:
is weakly ℱ-exhaustive at x;
f is continuous at x.
Proof (i) ⇒ (ii) Let be an -sequence in Y associated with weak ℱ-exhaustiveness of at x, and pick . By hypothesis, there exists a neighborhood of x, related with weak ℱ-exhaustiveness. Fix arbitrarily . There is a set (depending on x and z) with for all . Moreover, thanks to -convergence with respect to the -sequence , there exists a set (depending on x and z) with and whenever . Thus for every we get
⇒ (i) Since f is continuous at x, there exists an -sequence such that for each there is a neighborhood of x with(1)
By -convergence of to f on X with respect to the -sequence , there is a set with
for each . From (1) and (2) we obtain
for every . From (3) we get the existence of an -sequence with the property that for every there is a set (depending on x and z) with whenever . Thus the net is weakly ℱ-exhaustive at x. This ends the proof. □
Analogously as Theorem 3.1, it is possible to prove the following.
Theorem 3.2 Under the same notations and assumptions as in Theorem 3.1, suppose that , , -converges to on X with respect to a single -sequence in Y. Then the following are equivalent:
is weakly ℱ-exhaustive on X;
f is globally continuous on X.
We now use Theorem 3.2 to show that, in general, ℱ-exhaustiveness is strictly stronger than weak ℱ-exhaustiveness.
Example 3.3 Let ℱ be any fixed free filter of ℕ, be endowed with the usual distance, Σ be the σ-algebra of all Borel subsets of X, ν be the Lebesgue measure on X, and R be the space of all bounded ν-measurable functions on X, with identification up to ν-null sets. Note that order convergence in R coincides with almost everywhere convergence dominated by an element of R, which does not have a topological nature (see also ). For each real number a, let the function which associates to every element the constant a.
We consider the sequence , , defined as follows:
It is easy to see that -converges pointwise to with respect to an -sequence independent of (for example, , ) and thus, by Theorem 3.2, is weakly ℱ-exhaustive on . On the other hand, it is not hard to see that, if is any -sequence in R, then there is a positive real number with for every and . In correspondence with , for each set there is an integer , . So we get , and hence . From this it follows that is not ℱ-exhaustive at 0. Moreover, it is not difficult to check that is ℱ-backward exhaustive at 0, but not ℱ-forward exhaustive at 0.
The following two propositions extend [, Proposition 3.5] and will be useful in the sequel.
Proposition 3.4 Let ℱ, X, R be as in Theorem 3.2, , , be a function net, ℱ-exhaustive on X and -convergent to on X.
Then f is globally continuous on X, and the net -converges on every compact subset with respect to a single -sequence, independent of C.
Proof Global continuity of f on X follows from Theorem 3.2.
Let now , , be three -sequences in Y, related with ℱ-exhaustiveness of , -convergence on X and global continuity of f on X respectively, let be any compact set, and choose arbitrarily and . As is ℱ-exhaustive at x and f is globally continuous, there exist and an open neighborhood of x, with
for each and . By compactness of C there is a finite family , with for every , whose union contains C. Since -converges to f on X, in correspondence with and there is a set with
Let : note that . Pick arbitrarily : there exists at least with . Then from (4) and (5) we get
for each . This ends the proof. □
Proposition 3.5 Let ℱ, X, R be as above. If , , is a net of functions, globally continuous with respect to a single -sequence independent of λ and -convergent to on X, then f is globally continuous and is ℱ-exhaustive on X.
Proof We begin with proving global continuity of f on X. Let , and be two -sequences, related with -convergence of to f and global continuity of the ’s, respectively, and fix arbitrarily and . By hypothesis there is a set with for each and . Fix . By global continuity of there are a neighborhood U of x with for each . Thus for such z’s we get
namely global continuity of f with respect to the -sequence , where , .
We now prove ℱ-exhaustiveness of the net on X. Choose arbitrarily . By global continuity of f with respect to , in correspondence with and there is a neighborhood of x with whenever . By -convergence of to f on X with respect to , there is with
From (6) and (7) we get
for every and . This ends the proof. □
Remark 3.6 Proceeding analogously as above, it is possible to see that Theorems 3.1, 3.2 and Propositions 3.4, 3.5 hold even when the distance function ρ does not satisfy necessarily symmetric property, and - (-) convergence, (weak) ℱ-exhaustiveness and continuity are replaced by - (-) forward (backward) convergence, (weak) ℱ-forward (backward) exhaustiveness and forward (backward) continuity, respectively, under the hypothesis that the forward and backward convergences are equivalent.
Example 3.7 In general, the hypothesis that forward convergence implies backward convergence is essential. Indeed, let be endowed with the usual distance, be endowed with the following distance function:
Observe that is an asymmetric metric space (see also [, Example 5.10]). Let , , be defined by setting , , . Note that and . Thus, it is not difficult to check that for any free filter ℱ of ℕ the sequence ℱ-forward converges uniformly on to the function defined by , and that the ℱ-forward limit is unique, while the ℱ-backward limit of does not exist in R for any . Moreover, since for each , we find that f is neither forward nor backward continuous at 0. Furthermore, as
for every , it follows that for any free filter ℱ of ℕ the sequence is both ℱ-forward and ℱ-backward exhaustive at 0.
We now give some versions of Ascoli-type theorems in the context of lattice groups and filter exhaustive nets, extending earlier results proved in  and . Note that in our context, since we deal with abstract structures which are not necessarily by a topology, it will be advisable to deal with suitable notions of ‘filter closedness’ and ‘filter compactness’ in relation with convergences, which are not necessarily generated by a Hausdorff topology. For example, note that in the space of all measurable functions on with respect to the σ-algebra Σ of all Borel subsets of and the Lebesgue measure ν, with identification up to ν-null sets, order convergence coincides with almost everywhere convergence, which does not have a topological nature. Moreover, there exist Dedekind complete vector lattices which do not have any Hausdorff compatible vector topology, for which every bounded monotone increasing sequence converges to its supremum (see for instance ).
Given a directed set Λ, a -free filter ℱ of Λ, a topological space X, a cone metric space R and a nonempty set , we say that Φ is -compact (resp. -compact) iff every net in Φ admits a subnet , -convergent to an element (resp. -convergent to an element on every compact subset with respect to a single -sequence independent of C). We say that Φ is -closed iff whenever is a net in Φ, -convergent to . The -closure of Φ is the set of the functions , having a net in Φ, -convergent to f. Analogously as above, it is possible to formulate the notions of -closedness and of -closure. Note that Φ is -closed (resp. -closed) if and only if it coincides with its -closure (resp. -closure).
When , -convergence coincides with ℱ-pointwise convergence, and hence we denote the related above concepts by ℱ-compactness, ℱ-closedness and ℱ-closure, respectively.
We now are in a position to give the following abstract Ascoli-type theorem.
Theorem 3.8 Under the same notations and hypotheses as above, if , where Φ is -closed and Ψ is -compact, and
(H′) every -convergent net in Φ has a subnet , -convergent (in ) and ℱ-exhaustive on X,
then Φ is -compact.
Moreover, if Φ is -compact, then Φ satisfies condition (H′).
Proof We begin with the first part. Let be -closed, and be a net in Φ. Since and Ψ is -compact, has a -convergent subnet . By condition (H′), this subnet has a sub-subnet , -convergent to a function and ℱ-exhaustive on X. By Proposition 3.4, the net ℱ-converges to f uniformly on every compact subset , with respect to an -sequence independent of C. Since Φ is -closed, then . Therefore, Φ is -compact.
We now turn to the last part. Let Φ be -compact and be a -convergent net in Φ. Then it admits a subnet , -convergent to a function on every compact subset with respect to an -sequence, independent of C. By Proposition 3.5, we find that -converges to f and is ℱ-exhaustive on every compact set C with respect to a single -sequence, and hence, by arbitrariness of C, it -converges to f and is ℱ-exhaustive on X. This ends the proof. □
Remark 3.9 Observe that, in general, condition (H′) is strictly weaker than equicontinuity (see also [, Remark 3.2.17]).
A consequence of Theorem 3.8 is the following (see also ).
Theorem 3.10 Let and be asymmetric metric spaces, such that and are real-valued distance functions, be a fixed element of R, ℱ be any free filter of ℕ and suppose that
(3.10.1) each subset of R, ℱ-closed and ℱ-forward bounded with respect to , is ℱ-compact.
Let be such that
(3.10.2) every sequence in Φ, pointwise convergent in , has a -exhaustive subsequence;
(3.10.3) every sequence in Φ has a subsequence , ℱ-pointwise forward bounded in R with respect to .
Then every sequence in Φ admits a subsequence, uniformly convergent on every compact subset in the usual sense.
Proof Choose arbitrarily , set , and let be the closure of in R. We claim that is compact in R. Indeed, let . There exists a sequence in , convergent to y with respect to . So, in correspondence with there is a natural number with whenever . Moreover, there is a sequence in Φ such that for each . By (3.10.3), there exist a subsequence of , a positive real number and a set with whenever . Thus there exists with
getting forward boundedness of and hence ℱ-compactness of , thanks to the hypotheses. This implies that is also compact: indeed, by proceeding analogously as in [, Proposition 2.4], it is possible to prove that every ℱ-convergent sequence in an asymmetric metric space has a subsequence, convergent in the usual sense. As is compact in R, then, by virtue of the Tychonoff theorem, the set is compact in with respect to the pointwise convergence. Since , then we find that every sequence in Φ has a subsequence , pointwise convergent to a suitable function . From (3.10.2) and Theorem 3.8 used with , and , where the roles of Φ and Ψ are played by the -closure of Φ and respectively, we get the assertion. □
Corollary 3.11 Under the same hypotheses and notations as in Theorem 3.10, let ℱ be a P-filter of ℕ. Let satisfy (3.10.1), (3.10.2) and be such that
(3.11.1) every sequence in Φ, pointwise convergent in , has a ℱ-exhaustive subsequence.
Then every sequence has a subsequence, uniformly convergent on X in the usual sense.
Proof Let be any function sequence in Φ. Arguing analogously as in the proof of Theorem 3.10, we see that has a subsequence , pointwise convergent in the usual sense to a function . By (3.11.1), has a ℱ-forward exhaustive subsequence, say . Since ℱ is a P-filter, proceeding analogously as in [, Lemma 3.6], it is possible to see that has a -forward exhaustive subsequence. The assertion follows from Theorem 3.10. □
Example 3.12 In general, in Theorem 3.10, the condition (3.10.1) cannot be dropped. Indeed, let X, and , , be as in Example 3.7. Note that is bounded and closed (with respect to itself), and also ℱ-closed and ℱ-bounded for any free filter ℱ of ℕ. Let : since whenever and for every , and , it is not difficult to see that, for each free filter ℱ of ℕ, every subsequence of does not admit ℱ-forward limit as . From this it follows that is not ℱ-forward compact and hence it does not fulfil condition (3.10.1). Let now : it is not hard to check that the conditions (3.10.2) and (3.10.3) are satisfied. Note that, as seen in Example 3.7, the sequence does not have any subsequence, uniformly (backward) convergent in the usual sense, and so the thesis of Theorem 3.10 is false.
We now give some versions of abstract Ascoli-type theorems with respect to Lipschitz-type metrics. For a related literature, we refer to  and the bibliography therein.
Let be a metric space endowed with a real-valued distance function, R be a Dedekind complete vector lattice, endowed with the absolute value, and let us add to R an extra element +∞, satisfying the properties analogous to those of the element +∞ of the extended real line. We say that is Lipschitz iff there is a positive element with whenever , and in this case we set
If is not Lipschitz, then we put . Note that, even if X is a compact metric space, and is continuous, it may happen that : indeed, it is enough to take , .
We now fix a point and consider the following extended metric:
Given a directed set and a -free filter ℱ of Λ, we say that
iff there is an -sequence , with
In this case, we say that the net -converges to f. The net is -Cauchy iff there is an -sequence in R with the property that for every there is with whenever .
It is easy to check that every -convergent net is -Cauchy. We will prove also the converse implication. To this aim, we first give the following extension of [, Proposition 3.5] to the setting of filters and vector lattices.
Proposition 3.13 Let , , be a function net, -convergent to f, and be related with . Then, for every , -converges to f on the set .
Proof By hypothesis, there is an -sequence such that for every there is a set with
for every and . From (10) we get
whenever and . This ends the proof. □
Example 3.14 Observe that, in general, -convergence is strictly stronger than -convergence on compact sets. Indeed, let be a compact metric space, fix a point , and suppose that there exists at least a sequence in X, convergent to x with respect to d in the usual sense. Let R be any Dedekind complete vector lattice, u be a strictly positive element of R, and put , for every and . It is not difficult to check that for any and . From this it follows that the sequence -converges on X for every free filter ℱ of ℕ. On the other hand, we get
for every . This implies that, for any free filter ℱ of ℕ, the sequence does not converge to 0. Otherwise there exists an -sequence in R such that for every , and so for each there is with . By arbitrariness of p it follows that , but this is impossible. Thus does not -converge to 0, and hence is not -convergent to f (see also [, Example 3.1]).
As a consequence of Proposition 3.13, we prove the following completeness result, which extends [, Proposition 3.3].
Proposition 3.15 Under the same above notations and hypotheses, let , , be an -Cauchy net of functions, globally continuous with respect to a single -sequence. Then -converges to a globally continuous function .
Proof Let , , be a -Cauchy net, and be related with . Then the net is -Cauchy, and so it is -convergent, since R is Dedekind complete (see also ). Moreover, for every and we get
From (11), the hypotheses and since the net is -Cauchy, it follows that for every the net is -Cauchy, and so -convergent. For each , set .
We now prove that . Choose arbitrarily . Since by hypothesis is -Cauchy, there is with , and hence
From (12), since for every and , it follows that
From Propositions 3.5, 3.13 and (13) we get global continuity of f. This ends the proof. □
The next step is to give an Ascoli-type theorem involving . We say that a net , , is ℱ-finitely -bounded iff there exists a finite number q of globally continuous functions , of elements , of sets with , and a set such that for every and whenever .
Theorem 3.16 Let X be a metric space. If , where Φ is -closed and Ψ is -compact, and if we assume that
(3.16.1) every -convergent net in Φ is ℱ-finitely -bounded,
then Φ is -compact.
Proof Choose arbitrarily a net . By -compactness of Ψ, has a subnet , -convergent to a function . By (H′), is ℱ-finitely -bounded. Pick arbitrarily and let q, F and , , , , be according to ℱ-finite -boundedness of . Since the ’s are globally continuous, we find an -sequence such that for each there is a neighborhood of x with
If and , then from (14) we get
getting ℱ-exhaustiveness of . By Proposition 3.4, the net - converges to f on every compact subset with respect to a single -sequence, independent of C. This ends the proof. □
When and , it is possible to prove the following extension of [, Theorem 5.1].
Theorem 3.17 Let ℱ be any free filter of ℕ, X be a separable metric space, be -closed, and such that every sequence , , in Φ is ℱ-finitely -bounded. Then Φ is -compact.
Proof Let be an ℱ-finitely -bounded sequence, be related to , and F, q, , , , , be associated with ℱ-finite -boundedness of . By arguing analogously as in the proof of Theorem 3.16, formula (15), it is possible to show that the sequence , , satisfies condition (3.10.1) with respect to the filter . To prove (3.10.2), observe that
From (16) it follows that
for each and . Thus the sequence , , satisfies (3.10.2) with respect to . By Theorem 3.10, the sequence , , and so even the sequence , , has a subsequence, uniformly convergent to f on every compact subset in the usual sense. By -closedness of Φ, it follows that . This ends the proof. □
Example 3.18 Observe that, in Theorem 3.17, in general the hypothesis of ℱ-finite -boundedness cannot be dropped. Indeed, let , , be a function sequence, such that , and is linear in and in for each . It is not difficult to check that for every free filter ℱ of ℕ, for any , for each and for any choice of real numbers , of functions , continuous on , and of sets with , there exists at least an index such that is infinite: otherwise F should be finite, but this is impossible, since ℱ is a free filter of ℕ. Thus there is a sufficiently large integer , with
Thus, for any free filter ℱ of ℕ, the sequence is not ℱ-finitely -bounded. Furthermore, it is not hard to see that the sequence has no subsequence convergent uniformly on , and so the set is -closed, but not -compact (see also [, Example 5.3]).
Remark 3.19 The results here obtained hold, even if Y is a weakly σ-distributive lattice group endowed with -convergence (see for instance ).
Prove similar results on (weak) filter exhaustiveness in other abstract structures.
Find further versions of Ascoli-type theorems, by considering different kinds of convergences in various kinds of abstract spaces.
We have first given, for cone metric space function nets, a characterization of continuity of the limit function in terms of filter weak exhaustiveness and some relation between filter exhaustiveness and uniform convergence (on compact sets). These relations have been used in proving our main abstract Ascoli-type theorem. Indeed, the condition on existence of a (filter pointwise convergent with respect to a single order sequence and) filter exhaustive subnet is necessary and sufficient for compactness with respect to the uniform convergence of compact sets, given the compactness with respect to the pointwise convergence. In the classical case, it is possible to treat these subjects by dealing with the compact-open topology (which corresponds to convergence uniformly on compacta; see also ). But in a general Dedekind complete lattice group it may happen the non-existence of Hausdorff topologies ‘compatible’ with the order (see also ), and we need an approach different from the classical one, by considering concepts like closedness and compactness by means of nets (or sequences) of functions rather than by topologies. In Theorem 3.10 and Corollary 3.11, we have extended earlier results proved in  and , including both the symmetric and the asymmetric case.
Furthermore, we have dealt with Lipschitz-type metrics and considered also extended distance-type functions, which are related with not necessarily Lipschitz functions. The main Ascoli-type results given in  have been extended in the lattice group setting and in the context of filter convergence and exhaustiveness (and also a concept similar to ‘total boundedness’ has been presented in terms of filters), giving some compactness result again with respect to uniform convergence on compact sets. We have given also some examples to support the results proved in our setting.
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Our thanks to the referees for their remarks, which improved the exposition of the paper.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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Boccuto, A., Dimitriou, X. Ascoli-type theorems in the cone metric space setting. J Inequal Appl 2014, 420 (2014). https://doi.org/10.1186/1029-242X-2014-420
- lattice group
- cone metric space
- (free) filter
- global continuity
- filter convergence
- (weak) filter exhaustiveness
- Ascoli theorem
- Lipschitz metric