Abstract

P. Das et al. recently introduced and studied the notions of strong -summability with respect to an Orlicz function and -statistical convergence, where is a nonnegative regular matrix and is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing with a family of matrices and with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal has a countable base), continuing some of the author's previous work.

1. Introduction

Let us begin by recalling that an ideal on a nonempty set is a nonempty set of subsets of such that and is closed under the formation of subsets and finite unions. The ideal is called admissible if for each . For example, if is infinite, then the set of all finite subsets of forms an ideal on . If is an ideal, then is a filter on .

Now if is a sequence in a topological space and is an ideal on the set of natural numbers, then is said to be -convergent to if for every neighbourhood of the set belongs to (equivalently, ). In a Hausdorff space the -limit is unique if it exists. It will be denoted by . If is the ideal of all finite subsets of , then -convergence is equivalent to the usual convergence. Thus if is admissible, the usual convergence implies -convergence. For a normed space the set of all -convergent sequences in is a subspace of and the map is linear. We refer the reader to [1–4] for more information on -convergence.

Recall now that for a given infinite matrix with real or complex entries a sequence of (real or complex) numbers is said to be -summable to the number provided that each of the series is convergent and .

The matrix is called regular if every sequence that is convergent in the ordinary sense is also -summable to the same limit. A well-known theorem of Toeplitz states that is regular if and only if the following holds:(i),(ii),(iii).

Let us suppose for the moment that is regular and also nonnegative (i.e., for all ). We will denote by the set for every . Then is said to be -statistically convergent to if for every we have , where the symbol denotes the characteristic function of the set . If one takes to be the Cesàro matrix (i.e., for and for ) one gets the usual notion of statistical convergence as it was introduced by Fast in [5]. Note that the set of all subsets for which holds is an ideal on and -statistical convergence is nothing but convergence with respect to this ideal.

For any number the sequence is said to be strongly --summable to provided that for all and . The strong --summability is a linear consistent summability method and the strong --limit is uniquely determined whenever it exists. In [6] Connor proved that is statistically convergent to whenever it is strongly -Cesàro convergent to and the converse is true if is bounded. Practically the same proof given in [6] still works if one replaces the Cesàro matrix by an arbitrary nonnegative regular matrix . In particular, strong --summability and -statistical convergence are equivalent on bounded sequences (see also [7, Theorem 8]). More information on strong matrix summability can be found in [8] (for the case ) or [9].

In [10] Maddox proposed a generalisation of strong --summability by replacing the number with a sequence of positive numbers: the sequence is strongly --summable to if for every and .

Next, let us recall that a function is called an Orlicz function if it is increasing, continuous, and convex and satisfies as well as if and only if . If we drop the convexity and replace it by the condition for all , then is called a modulus. For example, the function defined by is an Orlicz function for and a modulus for . We will denote the set of all Orlicz functions by and the set of all moduli by .

Connor introduced another generalisation of strong matrix summability in [7]: if is a modulus, then is said to be strongly -summable to the limit with respect to if for all and . It is shown in [7, Theorem 8] that strong -summability with respect to implies -statistical convergence and that the converse holds for bounded sequences. In [11] Demirci replaced the modulus in Connor’s definition by an Orlicz function and studied which results carry over to this setting.

Another common generalised convergence method is that of almost convergence introduced by Lorentz in [12]. For this we first recall that a Banach limit is a linear functional on the space of all bounded real-valued sequences such that is shift-invariant (i.e., ), positive (i.e., if for all ), and fulfills . The existence of a Banach limit can be easily proved by means of the Hahn-Banach extension theorem. A sequence is said to be almost convergent to if for every Banach limit .

It is proved in [12] that almost convergence is equivalent to “uniform Cesàro convergence.” More precisely, a bounded sequence in is almost convergent to if and only if the following holds: where .

Lorentz subsequently introduced and studied the notion of -convergence by replacing the Cesàro matrix with an arbitrary real-valued regular matrix : a bounded sequence in is said to be -convergent to provided that Stieglitz further generalised the notion of almost convergence in the following way (cf. [13]): consider a sequence of matrices with entries in or and a bounded sequence of real or complex numbers. Then is said to be -convergent to the number if each of the series with is convergent and To obtain -convergence, take for and for .

Maddox introduced the -analogue of strong matrix summability in [14]. If each of the matrices is nonnegative and is a (not necessarily bounded) sequence in or , then is said to be strongly -convergent to provided that Very recently, the authors of [15] introduced the following definitions, combining matrices and ideals.

Definition 1 (cf. [15]). Let be a nonnegative regular matrix, an ideal on , and an Orlicz function. Let be any real or complex number. A sequence in or is said to be (i)strongly -summable to with respect to if (ii)-statistically convergent to if  for every .

It is proved in [15, Theorem 2.5] that -summability with respect to implies -statistical convergence (to the same limit) and the converse holds if the sequence is bounded and satisfies the -condition (i.e., there is a constant such that for all ).

We would like to propose here the following three definitions that include all the above mentioned generalised convergence methods.

First we define a sequence of functions from a set into a generalised metric space (same as a metric space except that is allowed to take values in ; for example, for , for all , and defines a generalised metric on ) to be uniformly convergent to the function along the ideal if for every there is some such that for every or, equivalently, for every , we have If , this yields the usual definition of uniform convergence. Also, this definition is a direct generalisation of the definition of -statistical uniform convergence given in [16]. The uniform convergence of to along clearly implies for all .

Now we come to the main definition.

Definition 2. Let be an ideal on and any nonempty set. Let be a family of (not necessarily regular) matrices with entries in or and a family in . Suppose that there is some such that Finally, let be a sequence in or and or . (i) is said to be -summable to provided that each of the series is convergent and (ii)If each matrix is nonnegative, then is said to be strongly -summable to with respect to if (iii)If each is nonnegative, then is said to be -statistically convergent to provided that for every

If for all in (ii) we simply speak of strong -summability. Clearly, strong -summability to implies -summability to provided that is bounded, for all and

Taking and for each and in (ii) and (iii) yields the definitions of strong -summability with respect to and of -statistical convergence. If we take and in (i) and (ii) we obtain the definitions of - and strong -convergence. Setting , for every and for all in (ii) gives us the definition of Maddox's strong --summability.

Note also that if each is nonnegative, then the set of all subsets , such that is an ideal on (the condition ensures ). The -statistical convergence is nothing but the convergence with respect to . In the case that is the infinite unit matrix for each we have .

In the next section we will start to investigate the above convergence methods.

2. Some Convergence Theorems

If not otherwise stated, we will denote by an ideal on , by a family of real or complex matrices (where is any nonempty index set) such that there is some with , and by a family in . Finally, denotes a sequence in or and an element of or , as in the previous section.

The following two propositions (wherein each is implicitly assumed to be nonnegative) generalise the aforementioned results from [15, Theorem 2.5]. The techniques used there followed the line of [17] while we will adopt the techniques from [6].

Proposition 3. Suppose that is strongly -summable to with respect to and that Then is also -statistically convergent to .

Proof. Let be arbitrary. By assumption there is some such that for all But we have for all . Hence for every and the proof is finished.

Proposition 4. Suppose that is bounded and -statistically convergent to . If is equicontinuous at and there exists an such that as well as then is also strongly -summable to with respect to .

Proof. Let be arbitrary. Take with . Since is equicontinuous at , we can find a such that for all and all .
Because is -statistically convergent to there is some such that for every It follows that for every and all and we are done.

So in particular, if and meet the requirements of both Propositions 3 and 4, then -statistical convergence and strong -summability with respect to coincide on bounded sequences. Note that all the assumptions on are satisfied if for a family of positive numbers which is bounded and bounded away from zero.

If , in other words, if then -convergence implies -statistical convergence (to the same limit). Thus if and additionally satisfy the requirements of Proposition 4, then for bounded sequences -convergence also implies strong -summability to the same limit. Concerning the consistency of ordinary -summability, we have the following sufficient conditions which are analogous to those of Toeplitz's theorem. We write for the set of all bounded sequences for which .

Lemma 5. Suppose that for all and Then for every bounded sequence in or , if , then is also -summable to .

Proof. Because of (25) we may assume . Let be arbitrary. Since , we have and hence by (24) there is some such that But for all and all and thus and we are done.

The next proposition is the direct generalisation of [18, Theorem 3.3] to our setting. Its proof is easy and moreover virtually the same as in [18] so it will be omitted.

Proposition 6. Suppose that we are given two families of nonnegative matrices and . If then .

In [19] it was proved that a bounded (real) sequence is statistically convergent to if and only if is Cesàro-summable to and the “variance” converges to . The proposition below is a generalisation of this result. We will use the notation provided that each is nonnegative.

First we need the following lemma, whose proof is analogous to those of Propositions 3 and 4 and will therefore be omitted.

Lemma 7. Suppose that and fulfill the requirements of Propositions 3 and 4 and let be a family in or . Put for all and . Then implies that for every and the converse is true if is bounded and for some .

Proposition 8. Let be bounded. Under the same hypotheses as in the previous lemma and the additional assumption that for all and is -statistically convergent to the number if and only if is -summable to and converges to along uniformly in .

Proof. In view of Lemma 7 it is enough to consider the case for all . We first assume that is -summable to and that Because of where and are as in Proposition 4, it follows that is strongly -summable to and hence by Proposition 3 it is also -statistically convergent to .
Conversely, let be -statistically convergent to . Then by Proposition 4 is also strongly -summable to and because of assumption (33) it follows that is -summable to . Moreover, we have and hence converges to along uniformly in .

According to [12, Theorem 2], for any regular matrix the -convergence of a sequence implies its almost convergence to the same limit and by [12, Theorem 3] the converse is true if satisfies . The following two results are generalisations of these facts. Their proofs remain virtually the same and will not be given here.

Proposition 9. Let be an infinite matrix in such that for all and . Put , where for and for .
Let be -summable to the value . Then is also almost convergent to .

Theorem 10. Let and be as in the previous proposition but assume additionally that for every , for some , and Let be the Cesàro matrix and suppose that the family arises from as from . Suppose further that the ideal is admissible and that is another ideal. Let be -summable to the value . Then is also -summable to .

In [4] the notion of -Cauchy sequences in arbitrary metric spaces, which generalises the notion of statistically Cauchy sequences of Fridy (cf. [20]), was introduced. A sequence in a metric space is said to be an -Cauchy sequence if for every there is some such that . For this yields just an equivalent formulation of the notion of an ordinary Cauchy sequence. Fridy’s notion of statistically Cauchy sequences is obtained by taking , where is the Cesàro matrix. It was proved in [4] that every -convergent sequence is -Cauchy (cf. [4, Proposition 1]) and that, in the case of an admissible ideal , the metric space is complete if and only if every -Cauchy sequence in is -convergent (cf. [4, Theorem 2]). The proof of [4, Theorem 2] also shows that every -convergent sequence possesses a subsequence which is convergent in the ordinary sense.

In [20] it was proved that a sequence of numbers is statistically convergent if and only if it is statistically Cauchy, but a third equivalent condition was obtained there as well; namely, a number sequence is statistically convergent if and only if there is a sequence which is convergent in the usual sense and coincides “almost everywhere” with , which in our notation means precisely .

It is clear that, for any two sequences in an arbitrary topological space, if is -convergent and , then is also -convergent. For the case of -statistical convergence of sequences of numbers we can prove a converse result provided that has a countable base that fulfills a certain condition with respect to the matrix-family . The proof uses the basic ideas from [20].

Theorem 11. Let be an admissible ideal with such that there is an increasing sequence in for which forms a base of and Then the sequence is -statistically convergent to if and only if there is a sequence which is -convergent to and fulfills .

Proof. We only have to show the necessity. So let be -statistically convergent to . Put and for every . Then for every there exists a set such that and by (38) we can find a strictly increasing sequence in such that Next we fix a strictly increasing sequence in such that for every . We write for . Then and .
Let for every and put It is easily checked that for every and hence is -convergent to .
Now it remains to show . To this end, fix and choose such that and .
Since , we can find with Then and for every and each we have and which completes the proof.

Note that condition (38) is in particular satisfied for if and each is a lower triangular matrix.

Making use of his aforementioned characterisation of statistical convergence, Fridy further proved in [20] the following Tauberian theorem for statistical convergence: a statistically convergent sequence which satisfies for is convergent in the ordinary sense. It is not too difficult to obtain the following slightly more general result by modifying the proof from [20] accordingly (there the functions , and below are simply and ). For the sake of brevity, we skip the details.

Theorem 12. Let be an admissible ideal and a lower triangular matrix such that and for every . Suppose that , and are functions from into itself such that is decreasing on , for every , whenever , and Let and be number sequences such that , , and for . Then .

Combining Theorems 11 and 12 we get the following corollary.

Corollary 13. Under the same general hypothesis as in Theorem 12 with , if is a sequence which is -statistically convergent to the number and fulfills for , then is convergent to in the usual sense.

3. Limit Superior and Limit Inferior

In [21] Demirci introduced the concepts of limit superior and limit inferior with respect to an ideal on , generalising the notions of statistical limit superior and limit inferior from [22]. For a sequence in put The same definitions were independently introduced by the authors of [3]. Note that since is not assumed to be bounded, it can happen that these values are or . If the above definitions are equivalent to the usual definitions of limit superior and limit inferior. It is proved in [21] (and in [3] as well) that and that is -convergent to if and only if (cf. [21, Theorems 3 and 4] or [3, Theorems 3.2 and 3.4]).

Let us also remark that as is easily checked.

In [22, Lemma on p.3628] necessary and sufficient conditions for a real matrix to satisfy the inequality for all were obtained (here, st-limsup  denotes the aforementioned statistical limit superior that was introduced in [22]; in our terminology it is nothing but the limit superior with respect to the ideal , where is the Cesàro matrix).

Later, Demirci gave a more general necessity result concerning the -limit superior and the -limit inferior (cf. [21, Corollary 1]). The following proposition is a further generalisation of this result while its proof follows the lines from [22].

Proposition 14. Let be ideals on and an infinite matrix in such that the following conditions are satisfied: Then as well as

Proof. Let be arbitrary and put . Since is bounded, we have . Also, fix an arbitrary . Then by [21, Theorem 1] (or [3, Theorem 3.1]) we have . We put .
For every set and , as in [22]. Note that and .
Then for every Because of and the assumptions (48) and (49) the -limit of the right-hand side of the above inequality is equal to . Together with the obvious monotonicity of - it follows that . Since was arbitrary, the proof is finished.
The second statement follows from the first one by multiplication with .

It was also proved in [22] that a sequence of real numbers which is bounded above and Cesàro-summable to its statistical limit superior is statistically convergent (cf. [22, Theorem 5]). It is possible to modify the proof of [22] to obtain the following more general result. We use the same notation as in the previous section.

Theorem 15. Suppose that each is nonnegative, for all , and If is a bounded sequence of real numbers and such that is -summable to and - or -, then is -statistically convergent to .

Proof. It is enough to prove the statement for the case -. Suppose that is not -statistically convergent to . Then - and hence there must be some such that . Consequently, there exists a such that Fix an arbitrary and put and . Take with . By our assumption (53) we have It follows from [21, Theorem 1] that and hence Now let be arbitrary. Since , there is some such that . Write . It then follows from the definitions of the sets , and and the choice of that Thus we have Suppose that Then it would follow that . But and hence contradicting (54).
Thus and since was arbitrary, we get and hence is not -summable to .

We conclude this section with a lemma that will be needed later and may also be of independent interest. First we need one more definition: a number sequence is called -bounded if there is a constant such that . Note that -convergent sequences are -bounded and that the -boundedness of implies that and are finite.

Lemma 16. For any ideal on and all -bounded sequences and in the inequalities hold. If one of the sequences is -convergent, then equality holds.

Proof. It is enough to prove the statement for the -. Let and . If such that and then and . Hence . But and thus .
If , then there would be some such that , which would imply . Thus we must have . Since and were arbitrary, it follows that .
Now suppose that is -convergent to and fix an arbitrary . Put , , and .
By [21, Theorem 1] and because of we have , that is, .
If , then and hence ; thus , contradicting the fact that .
So we must have and since , it follows that , which implies . Letting completes the proof.

4. Cluster Points

Fridy [23] defined and studied statistical cluster points and statistical limit points of a sequence. These concepts were later generalised by the authors of [1] to an arbitrary admissible ideal . Consider a sequence in a metric space . An element is called an -cluster point of if for every and it is called an -limit point of if there is a subsequence with that converges to . For , both notions are equivalent to the usual notion of cluster points. Every -limit point is also an -cluster point of (cf. [1, Proposition 4.1]) but the converse is not true in general. It was shown in [3, Theorem 3.5] that a bounded sequence in always possesses an -cluster point and that the - and the - of the sequence are the greatest and the smallest of them, respectively. It is easily observed that the same proof still works if the sequence is only -bounded.

Concerning -cluster points, we can give the following characterisation.

Proposition 17. Suppose that and Then is a -cluster point of if and only if for every

Proof. Put and for every . By definition, is a -cluster point of if and only if for every which is the case if and only if But , so because of (63) and Lemma 16 it follows that is a -cluster point of if and only if and the proof is finished.

This characterisation yields the following sufficient condition for a -cluster point.

Corollary 18. Under the same assumptions as in the previous proposition, if is a family in such that then is a -cluster point of .

Proof. For every and all we have and thus it follows from the assumptions that Hence by the previous proposition, is a -cluster point of .

5. Pre-Cauchy Sequences

The authors of [24] introduced the notion of statistically pre-Cauchy sequences. The sequence is called a statistically pre-Cauchy sequence if for every . They show that a statistically convergent sequence is statistically pre-Cauchy and that the converse is not true in general but under certain additional assumptions. It is further proved that is statistically pre-Cauchy if and that the converse is true if is bounded (cf. [24, Theorem 3]).

We propose the following generalisation of the definition of statistically pre-Cauchy sequences to our setting.

Definition 19. If each is nonnegative, a sequence of real or complex numbers is called a -statistically pre-Cauchy sequence if for every where .

First we show that, under an additional assumption on , -statistically convergent sequences are -statistically pre-Cauchy.

Lemma 20. Suppose that is -statistically convergent and Then is a -statistically pre-Cauchy sequence.

Proof. Say is -statistically convergent to . For every and all we have

The next two propositions are the analogues of [24, Theorem 3]. Since their proofs parallel very much those of Proposition 3 and Proposition 4, respectively, they will be omitted. In the formulation of both propositions, we differ from our usual notation and allow to be a family in with index set instead of .

Proposition 21. Suppose that Then is -statistically pre-Cauchy.

Proposition 22. Suppose that is bounded and -statistically pre-Cauchy. If is equicontinuous at and as well as then we also have

It was proved in [24] that a statistically pre-Cauchy sequence which possesses a convergent subsequence such that the set of indices is “large” in the sense that is statistically convergent. This result can be generalised in the following way.

Theorem 23. Suppose that and Let be any real or complex number. Let be a -statistically pre-Cauchy sequence and let be such that for every the set belongs to and furthermore Then is -statistically convergent to .

Proof. Take arbitrary. Then , by assumption. Put , , and . Then .
Let us also fix such that . Since is -statistically pre-Cauchy, there is some such that But we have and thus Since , it follows that Because of Lemma 16 this implies By [21, Theorem 2] we have If , then for every .
Thus with and the proof is finished.

By [24, Theorem 5] a bounded statistically pre-Cauchy sequence in whose set of cluster points is nowhere dense is statistically convergent. To obtain an analogous result in our setting, we introduce the following strengthening of the notion of -statistically pre-Cauchy sequences.

Definition 24. If each is nonnegative, a sequence of real or complex numbers is called a -statistically pre-Cauchy sequence if for every

For -statistically pre-Cauchy sequences, Lemma 20, Proposition 21, and Proposition 22 hold accordingly (with the obvious modifications, one can even take a family in with index set in this case).

The next lemma generalises [24, Lemma 4] while its proof follows the same lines.

Lemma 25. Let be an admissible ideal. Suppose that for all and Let be a basis for such that for every the following holds: Let be a -statistically pre-Cauchy sequence in and such that .
Then or .

Proof. Let us put if and if . Since , it is not difficult to see that is also -statistically pre-Cauchy. Put and . Then and ; thus it suffices to show or . Note also that for all and hence .
For the sake of brevity, we define for and We claim that To see this, fix an arbitrary and note that . So, since is -statistically pre-Cauchy, there is some such that By (90) there exists such that Because of (89) and this easily implies proving our claim. In particular, we can find with Then for every we must have Write and choose according to (91). Suppose first that . Then the same must hold for every ; for elsewise we could find a minimal with which would imply for all , contradicting the choice of .
So we have for all and all . Now fix again an arbitrary . By (93) there is such that Since is admissible, is again an element of and we have Thus we have shown that converges along to zero uniformly in , which means exactly that .
In the second case, , one can show analogously that .

Note that if and for all but finitely many , then we can take and condition (91) is satisfied. For the Cesàro matrix we even have .

As in [24], we can now use the above lemma to obtain a sufficient condition for -statistical convergence.

Theorem 26. Under the same general hypotheses as in the previous lemma, if is a -bounded -statistically pre-Cauchy sequence in such that the set of all -cluster points of is nowhere dense (note that is closed (cf. [1, Theorem 4.1(i)]), so “ nowhere dense” just means that has empty interior) in , then is -statistically convergent.

Proof. Suppose that is -bounded and -statistically pre-Cauchy but not -statistically convergent.
As mentioned before, the -boundedness assures that there is some . Since is not -statistically convergent there is an such that or . Without loss of generality, we assume the former.
As in [24], we will show that . If not, there would be an open interval such that .
It follows from Lemma 25 that or .
Since , we would have . But we can find with and because of the set cannot belong to where on the other hand it is contained in .
Thus has nonempty interior and the proof is finished.

As an immediate consequence of Theorem 26 we get the following corollary.

Corollary 27. Under the same general assumptions as in Lemma 25, if is a -statistically pre-Cauchy sequence in whose range is finite, then is -statistically convergent.

6. A Sup-Limsup-Theorem

In this section we will present the generalisation of Simons' equality that was announced in the abstract, but first we need to recall some definitions: a boundary for a real Banach space is a subset of (for every Banach space we denote by its closed unit ball and by its unit sphere) such that for every there is some with . By the Hahn-Banach-theorem, is always a boundary for . It easily follows from the Krein-Milman-theorem that ex , the set of extreme points of , is also a boundary for .

A famous theorem due to Rainwater (cf. [25]) states that a bounded sequence in which is convergent to some under every functional from ex is weakly convergent to .

Later Simons (cf. [26, 27]) generalised this result to an arbitrary boundary by proving that for every bounded sequence in the equality which is nowadays known as Simons’ equality, holds.

An easy separation argument shows that every boundary satisfies , but is not true in general (here denotes the convex hull, the weak-closure, and the norm-closure of ).

In [28] Fonf and Lindenstrauss introduced the following intermediate notion. Consider a convex weak-compact subset of (where is a real or complex Banach space). A subset of is said to -generate provided that whenever is written as a countable union , then or equivalently, whenever is written as a countable union with , then Clearly, implies that -generates which in turn implies , but the converses are not true in general as was shown in [28]. It was also proved in [28] that, for a real Banach space, every boundary of -generates    (the set is called a boundary of if for every . In this terminology, is a boundary for if and only if it is a boundary of ).

Nygaard proved in [29] that Rainwater’s theorem holds true for every -generating subset of and the authors of [30] showed that Simons’ equality is equivalent to the -generation property (cf. [30, Theorem 2.2]; see also [31, Lemma 2.1 and Remark 2.2]).

In [32] the author investigated the possibility to generalise the Rainwater-Simons-convergence theorem for -generating sets to some generalised convergence methods such as strong --summability and almost convergence by proving a general Simons-like inequality for -generating sets (cf. [32, Theorem 3.1]). We will continue this work here, using similar arguments as in [32] to generalise Simons' equality to the -limsup for the case that has a countable base, and obtain some related convergence results.

First we need the following lemma, whose proof is—once more—analogous to those of Propositions 3 and 4. Therefore, the details will be skipped.

Lemma 28. Let each be nonnegative. Define by for and for . Put for every . Then and the converse is true if the sequence is bounded and for some .

Now we turn to the generalisation of Simons’ equality.

Theorem 29. Let be a real Banach space, a convex weak*-compact subset, and an -generating set for . Let the ideal be such that the filter has a countable base. Assume that each is nonnegative and that there exists an such that Let be a bounded sequence in . Then the equality holds.

Proof. Denote the left-hand supremum by and the right-hand supremum by . We only have to show . Let . Let be a countable base for . Without loss of generality we may assume for all . Take and arbitrary and put where is as in the previous lemma. Then for every . It follows from [21, Theorem 1] that for every . Together with the previous lemma this easily implies .
Since -generates , we get that Thus we can find and with . It is easily checked that is convex and weak*-closed; hence . But for every It follows that for every and every . Since and was arbitrary, we conclude with Lemma 28 that for every , whence .

As a corollary, we get the following convergence result.

Corollary 30. Under the same hypotheses as in Theorem 29 with , if is such that is -statistically convergent to for every then the same holds true for every ; that is, is “weakly -statistically convergent to .”
Moreover, for every family in which is equicontinuous at and satisfies is strongly -summable to with respect to for every whenever this statement holds for every .

Proof. The first statement follows directly from Theorem 29 and the second follows from the first one via Propositions 3 and 4.

It is clear that this convergence result carries over to complex Banach spaces (note that if is a complex Banach space and -generates ; then -generates , the unit ball of the underlying real space).

In particular, if we take each to be the infinite unit matrix, we get that, for every ideal such that has a countable base, for every whenever this is true for every in an -generating subset of (in particular, in a boundary for ). We can also prove an analogous convergence result for -summability.

Proposition 31. Let be a real or complex Banach space and an -generating set for . Suppose that has a countable base, for all , and moreover for some .
Let be a bounded sequence in and such that is -summable to for every . Then the same is true for every .

Proof. Let be a decreasing countable basis for . Let and . Take any and fix an arbitrary . Define Then and since -generates , we can find and such that .
It is not too hard to see that is convex and weak*-closed and thus . Consequently, for all and we have Since and was arbitrary, we are done.

The next result concerning -statistically pre-Cauchy sequences is a generalisation of [32, Corollary 3.5]. Using Propositions 21 and 22 with for all and , its proof can be carried out analogously to that of Proposition 31. The details will be omitted.

Proposition 32. Let be a real or complex Banach space and an -generating set for . Suppose that has a countable base, that each is nonnegative, and that there is some such that Let be a bounded sequence in such that is -statistically pre-Cauchy and -statistically pre-Cauchy, respectively, for every . Then the same is true for every .

Finally, let us give characterisations of weak-compactness and reflexivity that generalise [32, Corollaries 3.7 and 3.8].

Corollary 33. Let be a bounded subset of the Banach space and an -generating set for . Then is weakly relatively compact if (and only if) for every sequence in there is an element , an ideal on such that admits a countable base, and a nonnegative matrix such that and is -statistically convergent to for every .

Proof. Let be an arbitrary sequence in and fix , , and as above. By Corollary 30 is -statistically convergent to for every . Thus, given finitely many functionals , the sequence is -statistically convergent to zero. Hence for any the set does not belong to .
By (119), is admissible; therefore, must be infinite for every , which shows that is a weak-cluster point of .
So is weakly relatively and countably compact and by the Eberlein-Shmulyan theorem, it must be also weakly relatively compact.

Corollary 34. If is an -generating set for (we consider canonically embedded into its bidual), then is reflexive if (and only if) for every sequence in there is a functional , an ideal on such that admits a countable base, and a nonnegative matrix such that (118) and (119) are satisfied and is -statistically convergent to for every .

Proof. By the previous corollary, is weakly compact; thus and hence also are reflexive.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is grateful to the anonymous referee for pointing out reference [16] and the final version of [19].