On convergence with respect to an ideal and a family of matrices

Recently P. Das, S. Dutta and E. Savas introduced and studied the notions of strong $A^I$-summability with respect to an Orlicz function $F$ and $A^I$-statistical convergence, where $A$ is a non-negative regular matrix and $I$ is an ideal on the set of natural numbers. In this note, we will generalise these notions by replacing $A$ with a family of matrices and $F$ 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 $I$ has a countable base), continuing the author's previous work.


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 F( ) := { \ : ∈ } is a filter on . Now if ( ) ∈N is a sequence in a topological space and is an ideal on the set N of natural numbers, then ( ) ∈N is said to be -convergent to ∈ if for every neighbourhood of the set { ∈ N : ∉ } belongs to (equivalently, { ∈ N : ∈ } ∈ F( )). In a Hausdorff space the -limit is unique if it exists. It will be denoted by -lim . If is the ideal of all finite subsets of N, 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 N and the map ( ) → -lim is linear. We refer the reader to [1][2][3][4] for more information on -convergence.
Recall now that for a given infinite matrix = ( ) , ∈N with real or complex entries a sequence = ( ) ∈N of (real or complex) numbers is said to be -summable to the number provided that each of the series ∑ ∞ =1 is convergent and lim → ∞ ∑ ∞ =1 = .
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: Let us suppose for the moment that is regular and also nonnegative (i.e., ≥ 0 for all , ∈ N). We will denote by ( , , ) the set { ∈ N : | − | ≥ } for every > 0. Then is said to be -statistically convergent to if for every > 0 we have lim → ∞ ∑ ∞ =1 ( , , ) ( ) = 0, where the symbol denotes the characteristic function of the set ⊆ N. If one takes to be the Cesàro matrix (i.e., = 1/ for ≤ and = 0 for > ) one gets the usual notion of statistical convergence as it was introduced by Fast in [5]. Note that the set of all subsets ⊆ N for which lim → ∞ ∑ ∞ =1 ( ) = 0 holds is an ideal on N and -statistical convergence is nothing but convergence with respect to this ideal. 2 International Journal of Analysis For any number > 0 the sequence is said to be strongly --summable to provided that ∑ ∞ =1 | − | < ∞ for all ∈ N and lim → ∞ ∑ ∞ =1 | − | = 0. 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 = 1) or [9].
In [10] Maddox proposed a generalisation of strong --summability by replacing the number with a sequence p = ( ) ∈N of positive numbers: the sequence is strongly -p-summable to if ∑ ∞ =1 | − | < ∞ for every ∈ N and lim → ∞ ∑ ∞ =1 | − | = 0. Next, let us recall that a function : [0, ∞) → [0, ∞) is called an Orlicz function if it is increasing, continuous, and convex and satisfies lim → ∞ ( ) = ∞ as well as ( ) = 0 if and only if = 0. If we drop the convexity and replace it by the condition ( + ) ≤ ( ) + ( ) for all , ≥ 0, then is called a modulus. For example, the function defined by ( ) = is an Orlicz function for ≥ 1 and a modulus for 0 < ≤ 1. We will denote the set of all Orlicz functions by O and the set of all moduli by M.
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 ∑ ∞ =1 (| − |) < ∞ for all ∈ N and lim → ∞ ∑ ∞ =1 (| − |) = 0. 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 ℓ ∞ of all bounded real-valued sequences such that is shift-invariant (i.e., (( +1 ) ∈N ) = (( ) ∈N )), positive (i.e., (( ) ∈N ) ≥ 0 if ≥ 0 for all ), and fulfills (1, 1, . . .) = 1. The existence of a Banach limit can be easily proved by means of the Hahn-Banach extension theorem. A sequence ∈ ℓ ∞ is said to be almost convergent to ∈ R if ( ) = for every Banach limit . It is proved in [12] that almost convergence is equivalent to "uniform Cesàro convergence. " More precisely, a bounded sequence = ( ) ∈N in R is almost convergent to ∈ R if and only if the following holds: Lorentz subsequently introduced and studied the notion of -convergence by replacing the Cesàro matrix with an arbitrary real-valued regular matrix : a bounded sequence = ( ) ∈N in R is said to be -convergent to ∈ R provided that Stieglitz further generalised the notion of almost convergence in the following way (cf. [13]): consider a sequence B = ( ) ∈N 0 = (( ( ) ) , ∈N ) ∈N 0 of matrices with entries in R or C and a bounded sequence = ( ) ∈N of real or complex numbers. Then is said to be B -convergent to the number if each of the series ∑ ∞ =1 ( ) with ∈ N, ∈ N 0 is convergent and To obtain -convergence, take ( ) = − for > and ( ) = 0 for ≤ .
Maddox introduced the B -analogue of strong matrix summability in [14]. If each of the matrices is nonnegative and = ( ) ∈N is a (not necessarily bounded) sequence in R or C, then is said to be strongly B -convergent to provided that Very recently, the authors of [15] introduced the following definitions, combining matrices and ideals.
Definition 1 (cf. [15]). Let = ( ) , ∈N be a nonnegative regular matrix, an ideal on N, and an Orlicz function. Let be any real or complex number. A sequence = ( ) ∈N in R or C is said to be (i) strongly -summable to with respect to if -lim for every > 0.
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 Δ 2 -condition (i.e., there is a constant such that (2 ) ≤ ( ) for all ≥ 0). We would like to propose here the following three definitions that include all the above mentioned generalised convergence methods.
International Journal of Analysis 3 First we define a sequence ( ) ∈N of functions from a set into a generalised metric space ( , ) (same as a metric space except that is allowed to take values in [0, ∞]; for example, ( , ) = | − | for , ∈ [0, ∞), ( , ∞) = (∞, ) = ∞ for all ∈ [0, ∞), and (∞, ∞) = 0 defines a generalised metric on [0, ∞]) to be uniformly convergent to the function : → along the ideal if for every > 0 there is some ∈ such that for every ∈ { ∈ N : ( ( ) , ( )) ≥ } ⊆ (7) or, equivalently, for every > 0, 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 ( ) ∈N to along clearly implies -lim ( ) = ( ) for all ∈ . Now we come to the main definition.

Definition 2.
Let be an ideal on N and any nonempty set. Let B = ( ) ∈ = (( ( ) ) , ∈N ) ∈ be a family of (not necessarily regular) matrices with entries in R or C and F = ( ( ) ) ∈N, ∈ a family in M ∪ O. Suppose that there is some Finally, let = ( ) ∈N be a sequence in R or C and ∈ R or C.
(i) is said to be B -summable to provided that each of the series ∑ ∞ =1 ( ) is convergent and (ii) If each matrix is nonnegative, then is said to be strongly B -summable to with respect to F if (iii) If each is nonnegative, then is said to be Bstatistically convergent to provided that for every If ( ) = id [0,∞) for all ∈ N, ∈ in (ii) we simply speak of strong B -summability. Clearly, strong B -summability to implies B -summability to provided that is bounded, Taking = and ( ) = ∈ O for each ∈ and ∈ N in (ii) and (iii) yields the definitions of strong -summability with respect to and of -statistical convergence. If we take = and = N 0 in (i) and (ii) we obtain the definitions of B -and strong B -convergence. Setting = , = for every ∈ and ( ) = for all ∈ , ∈ N in (ii) gives us the definition of Maddox's strong -p-summability.
Note also that if each is nonnegative, then the set B, of all subsets ⊆ N, such that is an ideal on N (the condition (+) ensures N ∉ B, ). The Bstatistical convergence is nothing but the convergence with respect to B, . In the case that is the infinite unit matrix for each ∈ we have B, = .
In the next section we will start to investigate the above convergence methods.

Some Convergence Theorems
If not otherwise stated, we will denote by an ideal on N, by B = ( ) ∈ = (( ( ) ) , ∈N ) ∈ a family of real or complex matrices (where is any nonempty index set) such that there is some 0 ∈ with (+), and by F = ( ( ) ) ∈N, ∈ a family in M ∪ O. Finally, = ( ) ∈N denotes a sequence in R or C and an element of R or C, 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 B -summable to with respect to F and that Then is also B -statistically convergent to .
Proof. Let , > 0 be arbitrary. By assumption there is some ∈ such that for all ∈ { ∈ N : But we have International Journal of Analysis for all ∈ , ∈ N. Hence for every ∈ and the proof is finished.

Proposition 4.
Suppose that is bounded and B -statistically convergent to . If F is equicontinuous at 0 and there exists an ∈ such that as well as then is also strongly -summable to with respect to F.
So in particular, if B and F meet the requirements of both Propositions 3 and 4, then B -statistical convergence and strong B -summability with respect to F coincide on bounded sequences. Note that all the assumptions on F are satisfied if ( ) = for a family ( ) ∈N, ∈ of positive numbers which is bounded and bounded away from zero.
If ⊆ B, , in other words, if then -convergence implies B -statistical convergence (to the same limit). Thus if B and F additionally satisfy the requirements of Proposition 4, then for bounded sequences -convergence also implies strong B -summability to the same limit. Concerning the consistency of ordinary Bsummability, we have the following sufficient conditions which are analogous to those of Toeplitz's theorem. We write for the set of all bounded sequences ( ) ∈N for which { ∈ N : ̸ = 0} ∈ .
Then for every bounded sequence Proof. Because of (25) we may assume = 0. Let > 0 be arbitrary. Since -lim = 0, we have := { ∈ N : | | ≥ } ∈ and hence by (24) there is some ∈ such that But for all ∈ and all ∈ N \ 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
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" 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 F and B fulfill the requirements of Propositions 3 and 4 and let
implies that for every > 0 and the converse is true if is bounded and sup ∈N, ∈N\ | | < ∞ for some ∈ .

Proposition 8. Let be bounded. Under the same hypotheses as in the previous lemma and the additional assumption that
is B -statistically convergent to the number if and only if is B -summable to and B,F i ( ) converges to 0 along uniformly in ∈ .
Proof. In view of Lemma 7 it is enough to consider the case = id [0,∞) for all ∈ N, ∈ . We first assume that is B -summable to and that where and are as in Proposition 4, it follows that is strongly B -summable to and hence by Proposition 3 it is also B -statistically convergent to .
Conversely, let be B -statistically convergent to . Then by Proposition 4 is also strongly B -summable to and because of assumption (33) it follows that is B -summable to . Moreover, we have and hence B,F ( ) converges to 0 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 following two results are generalisations of these facts. Their proofs remain virtually the same and will not be given here.
Let ∈ ℓ ∞ be A -summable to the value . Then is also almost convergent to .

Theorem 10. Let and A be as in the previous proposition but assume additionally that -lim
= 0 for every ∈ N, International Journal of Analysis Let be the Cesàro matrix and suppose that the family C arises from as A from . Suppose further that the ideal is admissible and that is another ideal. Let ∈ ℓ ∞ be Csummable to the value . Then is also A -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 ( ) ∈N in a metric space ( , ) is said to be an -Cauchy sequence if for every > 0 there is some ∈ N such that { ∈ N : ( , ) ≥ } ∈ . 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 ( ) ∈N is statistically convergent if and only if there is a sequence ( ) ∈N which is convergent in the usual sense and coincides "almost everywhere" with ( ) ∈N , which in our notation means precisely { ∈ N : It is clear that, for any two sequences ( ) ∈N , ( ) ∈N in an arbitrary topological space, if ( ) ∈N is -convergent and { ∈ N : ̸ = } ∈ , then ( ) ∈N is also -convergent. For the case of B -statistical convergence of sequences of numbers we can prove a converse result provided that F( ) has a countable base that fulfills a certain condition with respect to the matrix-family B. The proof uses the basic ideas from [20]. Theorem 11. Let be an admissible ideal with ⊆ B, such that there is an increasing sequence ( ) ∈N in for which {N\ : ∈ N} forms a base of F( ) and Then the sequence = ( ) ∈N is B -statistically convergent to if and only if there is a sequence ( ) ∈N which is -convergent to and fulfills { ∈ N : Proof. We only have to show the necessity. So let be Bstatistically convergent to . Put = 2 − and = { ∈ N : | − | ≥ } for every ∈ N. Then for every ∈ N there exists a set ∈ such that and by (38) we can find a strictly increasing sequence Next we fix a strictly increasing sequence ( ) ∈N in N such that ⊆ for every ∈ N. We write for . Then It is easily checked that { ∈ N : | − | ≥ } ⊆ for every and hence ( ) ∈N is -convergent to . Now it remains to show := { ∈ N : ̸ = } ∈ B, . To this end, fix > 0 and choose such that ∑ ∞ = +1 Then ∪ ∈ and for every ∈ N \ ( ∪ ) and each ∈ S we have ( ) > and which completes the proof.
Note that condition (38) is in particular satisfied for = {1, . . . , } if = and each is a lower triangular matrix.
Making use of his aforementioned characterisation of statistical convergence, Fridy further proved in [20] the International Journal of Analysis 7 following Tauberian theorem for statistical convergence: a statistically convergent sequence ( ) ∈N which satisfies | − +1 | = (1/ ) 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 ℎ below are simply ( ) = 1/ = ( ) and ℎ( ) = (1 + ) −1 ). For the sake of brevity, we skip the details.

Limit Superior and Limit Inferior
In [21] Demirci introduced the concepts of limit superior and limit inferior with respect to an ideal on N, generalising the notions of statistical limit superior and limit inferior from [22]. For a sequence ( ) ∈N in R put -lim sup := sup { ∈ R : { ∈ N : > } ∉ } , The same definitions were independently introduced by the authors of [3]. Note that since ( ) ∈N 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 -lim inf ≤ -lim sup and that ( ) ∈N is -convergent to ∈ R if and only if -lim inf = = -lim sup (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 lim sup ≤ st-lim sup for all ∈ ℓ ∞ 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].

International Journal of Analysis
Because of ∈ and the assumptions (48) and (49) thelimit of the right-hand side of the above inequality is equal to + . Together with the obvious monotonicity of -lim sup it follows that -lim sup ≤ + . Since > 0 was arbitrary, the proof is finished.
The second statement follows from the first one by multiplication with −1.
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. Proof. It is enough to prove the statement for the case B,lim sup = . Suppose that is not -statistically convergent to . Then B, -lim inf < and hence there must be some < such that := { ∈ N : < } ∉ B, . Consequently, there exists a > 0 such that Fix an arbitrary > 0 and put := { ∈ N : ≤ ≤ + } and := { ∈ N : > + }. Take ∈ (0, ) with | + | ≤ . By our assumption (53) we have It follows from [21, Theorem 1] that ∈ B, and hence Now let ∈ := ∩ (N \ ( ∪ )) be arbitrary. Since ∈ , there is some ∈ such that ∑ ∞ =1 ( ) ( ) > /2. Write = ‖ ‖ ∞ . It then follows from the definitions of the sets , , , , and and the choice of that Then it would follow that ∈ . But , ∈ and hence contradicting (54).
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 ( ) ∈N is called -bounded if there is a constant > 0 such that { ∈ N : | | > } ∈ . Note that -convergent sequences are -bounded and that the -boundedness of ( ) ∈N implies that -lim sup and -lim inf are finite. If ∩ ∈ , then (N \ ) ∪ (N \ ) ∈ F( ) and hence (N \ )∩ = ((N \ )∪(N \ )) ∩ ∈ F( ); thus N \ ∈ F( ), contradicting the fact that ∉ .

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 ( ) ∈N in a metric space ( , ). An element ∈ is called an -cluster point of ( ) ∈N if { ∈ N : ( , ) < } ∉ for every > 0 and it is called an -limit point of ( ) ∈N if there is a subsequence ( ) ∈N with { : ∈ N} ∉ that converges to . For = , both notions are equivalent to the usual notion of cluster points. Every -limit point is also an -cluster point of ( ) ∈N (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 ( ) ∈N in R always possesses an -cluster point and that thelim sup and the -lim inf 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 B, -cluster points, we can give the following characterisation. This characterisation yields the following sufficient condition for a B, -cluster point.

Corollary 18. Under the same assumptions as in the previous
then is a B, -cluster point of .
Proof. For every > 0 and all ∈ , ∈ N we have 10 International Journal of Analysis and thus it follows from the assumptions that Hence by the previous proposition, is a B, -cluster point of .

Pre-Cauchy Sequences
The authors of [24] introduced the notion of statistically pre-Cauchy sequences. The sequence = ( ) ∈N is called a statistically pre-Cauchy sequence if lim → ∞ 1/ 2 |{( , ) ∈ {1, . . . , } 2 : | − | ≥ }| = 0 for every > 0. 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.

Lemma 20. Suppose that is B -statistically convergent and
Then is a B -statistically pre-Cauchy sequence.
Proof. Say is B -statistically convergent to . For every > 0 and all ∈ N \ 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 F = ( ( ) ) , ∈N, ∈ to be a family in M ∪ O with index set N × N × instead of N × .
Then is B -statistically pre-Cauchy.

Proposition 22. Suppose that is bounded and
It was proved in [24] that a statistically pre-Cauchy sequence ( ) ∈N which possesses a convergent subsequence ( ) ∈N such that the set of indices { : ∈ N} is "large" in the sense that is statistically convergent. This result can be generalised in the following way. Let us also fix ∈ (0, ) such that ( − ) −1 ≤ . Since is B -statistically pre-Cauchy, there is some ∈ such that But we have and thus Since ∈ ⊆ B, , it follows that Because of Lemma 16 this implies By [21, Theorem 2] we have and the proof is finished.
By [24, Theorem 5] a bounded statistically pre-Cauchy sequence in R 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 B -statistically pre-Cauchy sequences.

Theorem 26.
Under the same general hypotheses as in the previous lemma, if = ( ) ∈N is a B, -bounded B + -statistically pre-Cauchy sequence in R such that the set of all B,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 R, then is B -statistically convergent.
Proof. Suppose that is B, -bounded and B + -statistically pre-Cauchy but not B -statistically convergent.
As mentioned before, the B, -boundedness assures that there is some ∈ . Since is not B -statistically convergent there is an > 0 such that { ∈ N : ≤ − } ∉ B, or { ∈ N : ≥ + } ∉ B, . Without loss of generality, we assume the former.
As in [24], we will show that ( Thus has nonempty interior and the proof is finished.
As an immediate consequence of Theorem 26 we get the following corollary. Lemma 25, if is a B + -statistically pre-Cauchy sequence in R whose range is finite, then is B -statistically convergent.

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 ( ) ∈N in the equality which is nowadays known as Simons' equality, holds. An easy separation argument shows that every boundary satisfies * = co * , but * = co is not true in general (here co 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 = ⋃ ∞ =1 , then co ( International Journal of Analysis 13 or equivalently, whenever is written as a countable union Clearly, = co implies that ( )-generates which in turn implies = co * , 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 max{ * ( ) : * ∈ } = sup{ * ( ) : * ∈ } 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. [ 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 -p-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 B, -limsup for the case that F( ) 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. 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 F( ) has a countable base. Assume that each is nonnegative and that there exists an ∈ such that Let ( ) ∈N be a bounded sequence in . Then the equality holds.
Proof. Denote the left-hand supremum by and the righthand supremum by . We only have to show ≤ . Let = sup ∈N ‖ ‖. Let ( ) ∈N be a countable base for F( ). Without loss of generality we may assume +1 ⊆ for all . Take * ∈ and > 0 arbitrary and put where is as in the previous lemma. Then ⊆ +1 for every ∈ N. It follows from [21, Theorem 1] that { ∈ N : * ( ) > + } ∈ B, for every > 0. Together with the previous lemma this easily implies ⋃ ∞ =1 = .
As a corollary, we get the following convergence result.

Corollary 30. Under the same hypotheses as in Theorem 29 with
= * , if ∈ is such that ( * ( )) ∈N is Bstatistically convergent to * ( ) for every * ∈ then the same holds true for every * ∈ * ; that is, ( ) ∈N is "weakly Bstatistically convergent to . " Moreover, for every family F = ( ( ) ) ∈N, ∈ in M ∪ O which is equicontinuous at 0 and satisfies ( * ( )) ∈N is strongly B -summable to * ( ) with respect to F 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 {Re * : * ∈ } ( )generates {Re * : * ∈ * }, 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 F( ) has a countable base, -lim * ( ) = * ( ) 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 B -summability. for some ∈ . Let ( ) ∈N be a bounded sequence in and ∈ such that ( * ( )) ∈N is B -summable to * ( ) for every * ∈ . Then the same is true for every * ∈ * .
It is not too hard to see that is convex and weak *closed and thus * ∈ . Consequently, for all ∈ and ∈ ∩ (N \ ) we have Since ∩ (N \ ) ∈ F( ) and > 0 was arbitrary, we are done.
The next result concerning B -statistically pre-Cauchy sequences is a generalisation of [32, Corollary 3.5]. Using Propositions 21 and 22 with ( ) = id [0,∞) for all , ∈ N 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 F( ) has a countable base, that each is nonnegative, and that there is some ∈ such that sup { ∞ ∑ =1 ( ) : ∈ N \ , ∈ } < ∞. (117) Let ( ) ∈N be a bounded sequence in such that ( * ( )) ∈N is B -statistically pre-Cauchy and B + -statistically pre-Cauchy, respectively, for every * ∈ . Then the same is true for every * ∈ * .

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 ( ) ∈N in there is an element ∈ , an ideal on N such that F( ) admits a countable base, and a nonnegative matrix = ( ) , ≥1 such that and ( * ( )) ∈N is -statistically convergent to * ( ) for every * ∈ .
By (119), , is admissible; therefore, must be infinite for every > 0, which shows that is a weak-cluster point of ( ) ∈N .
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 ( * ) ∈N in * there is a functional * ∈ * , an ideal on N such that F( ) admits a countable base, and a nonnegative matrix such that (118) and (119) are satisfied and ( * ( )) ∈N 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.