`International Journal of Mathematics and Mathematical SciencesVolume 2011, Article ID 517841, 10 pageshttp://dx.doi.org/10.1155/2011/517841`
Research Article

Statistical Convergence and Ideal Convergence of Sequences of Functions in 2-Normed Spaces

Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran 19569, Iran

Received 2 November 2010; Revised 22 February 2011; Accepted 12 March 2011

Abstract

We present various kinds of statistical convergence and -convergence for sequences of functions with values in 2-normed spaces and obtain a criterion for -convergence of sequences of functions in 2-normed spaces. We also define the notion of -equistatistically convergence and study -equi-statistically convergence of sequences of functions.

1. Introduction

The concept of ideal convergence was introduced first by Kostyrko et al. [1] as an interesting generalization of statistical convergence [25].

Throughout this paper will denote the set of positive integers. Let be a normed space. Let be a subset of positive integers and . The quotient is called the j’th partial density of and is a probability measure on , with support [2, 3].

The limit () (if exists) is called the natural density of . Clearly, finite subsets have natural density zero and where , that is, the complement of . If and have natural densities then . Furthermore, if , then [6].

Recall that a sequence of elements of is called to be statistically convergent to if the set has natural density zero for each . In this case we write - [24].

A family of subsets a nonempty set is said to be an ideal in if(i), (ii), (iii),

while an admissible ideal of further satisfies for each [7, 8]. Let be a nontrivial ideal in . The sequence in is said to be -convergent to , if for each the set = belongs to [1, 9].

2. Preliminaries

The notion of linear 2-normed spaces has been investigated by Gähler in the 60’s [10, 11] and this has been developed extensively in different subjects by others [1214]. Let be a real linear space of dimension greater than 1, and be a nonnegative real-valued function on satisfying the following conditions:(G1) if and only if and are linearly dependent vectors;(G2) for all in ;(G3) where is real,(G4) for all in

is called a 2-norm on and the pair is called a linear 2-normed space. In addition, for all scalars and all in , we have the following properties:(1) is nonnegative;(2); (3).

Some of the basic properties of 2-norm are introduced in [14]. Given a 2-normed space , one can derive a topology for it via the following definition of the limit of a sequence: a sequence in is said to be convergent to in if for every . This can be written by the formula We write it as

Lemma 2.1 (see [13]). Let be a basis of . A sequence in X is convergent to in if and only if for every . We can define the norm on by

Lemma 2.2 (see [13]). A sequence in is convergent to in if and only if .

Example 2.3. Let be equipped with the 2-norm := the area of the parallelogram spanned by the vectors and , which may be given explicitly by the formula Take the standard basis for .
Then, and , and so the derived norm with respect to is Thus, here the derived norm is exactly the same as the uniform norm on . Since the derived norm is a norm, it is equivalent to the Euclidean norm on .

Definition 2.4. Let be a nontrivial ideal in . The sequence of is said to be -convergent to , if for each and nonzero in the set belongs to [9].

If is -convergent to then we write it as The element is - of the sequence .

Remark 2.5. If is any sequence in and is any element of , then the set since if , so the above set is empty.

Further we will give some examples of ideals and corresponding -convergences.

Now we give an example of -convergence in 2-normed spaces.

Example 2.6. (i) Let be the family of all finite subsets of . Then is an admissible ideal in and -convergence coincides with usual convergence [11].
(ii) Put = . Then is an admissible ideal in and -convergence coincides with the statistical convergence [15].

Example 2.7. Let . Define the in 2-normed space by and let and . Then for every and we have that This implies that . But, the sequence is not convergent to .

3. Convergence for Sequences of Functions in 2-Normed Spaces

We discuss various kinds of convergence and -convergence for sequences of functions with values in 2-normed spaces.

Let be 2-normed spaces and assume that functions are given.

Definition 3.1. The sequence is said to be positive convergent to (on ) if We write

This can be expressed by the formula

Remark 3.2. If functions are given as in Definition 3.1 and dim then () is pointwise convergent to (on ) if and only if

We introduce uniform convergent of to by the formula and we write it as

Example 3.3. If is introduced in Lemma 2.1 then define then

Example 3.4. Let and define Then obviously . But we show that does not converge uniformly to in . Fix and for all put = then

Definition 3.5. Let and be 2-normed spaces with and let be a function. The function is said to be sequentially continuous at if for any sequence of converging to one has

Definition 3.6. Let and be two 2-normed spaces, and dimY . If is a sequence of functions, we say is equi-continuous (on ) if

Corollary 3.7. Let and be two 2-normed spaces, with dim.
If is a function such that satisfying the following formula then is sequentially continuous at .

Proof. Let be a sequence in such that . Let . There exists such that for every where for each . On the other hand hence for all there exist such that for all . Therefore and is sequentially continuous at .

4. -Convergence of Functions in 2-Normed Spaces

Let be 2-normed spaces. Fix an admissible ideal and assume that functions are given.

Definition 4.1. A sequence of functions is said to be -pointwise convergent to f  (on ) if - (in ) for each . We Write
This can be expressed by the formula

Definition 4.2. A sequence, is said to be -uniformly convergent to (on ) if and only if We write .

Remark 4.3. If then and will be read (respectively) as -pointwise and -uniform statistically convergence. If , then for all which may be given by the formula we have by [15]

Remark 4.4. We obviously have

Remark 4.5. Let be such that -convergence of sequences of points in is strictly more general than the usual convergence. Then there is a sequence , such that Putting and for and , we have Thus, in this situation, -uniform convergence of sequences of functions is strictly more general than the usual uniform convergence.

Theorem 4.6. Let be an admissible ideal and be two 2-normed spaces with dim. Assume that (on ) where functions are equi-continuous (on X) and . Then is sequentially continuous (on ).

Proof. Let and . By equi-continuty of ’s, there exist such that for every whenever .
Fix such that . Since , the set is in and different from . Hence there exists such that We have and by (Corollary 3.7) is sequentially continuous at .

5. -Equistatistically Convergent

Let be two 2-normed spaces with dim and be admissible ideal on .

Definition 5.1. A is called -equi-statistically convergent to (we write it as ) if for every the sequence of functions given by is uniformly convergent to the zero function (on ). Hence if and only if the following formula holds:

Corollary 5.2. The following properties hold:(i) implies ,(ii) implies .

Proof. (i)If by the monotonicity of operator , we take in Definition 4.2. Thus it is obvious.(ii)Assume and . By Definition 4.2 there exist a set such that for all and . Since . We can pick such that for all . Let and . Thus implies . Hence for each , we have by Definition 4.2 witnesses that .

Example 5.3. Define : Then but . Indeed, let and such that . Then for all and we have Hence .
Suppose that . Thus there is the set such that for all and we have .
Choose . Then must be the zero function, a contradiction.

Theorem 5.4. Assume and for fix . If and all functions , are sequentially continuous at then f is sequentially continuous at .

Proof. Let . Since , we can find such that Put . In other word is a probability measure on with the support , it follows that for all . By the sequentially continuity of at , there exist such that for all and for each . Fix such that , for each .
Since and , there exists such that
Thus we show that for all for each .

References

1. P. Kostyrko, M. Macaj, and T. Salat, “-Convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000.
2. H. Fast, “Sur la convergence statistique,” Colloqium Mathematicum, vol. 2, pp. 241–244, 1951.
3. J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
4. E. Kolk, “The statistical convergence in Banach spaces,” Tartu Ülikooli Toimetised, no. 928, pp. 41–52, 1991.
5. H. Steinhaus, “Sur la convergence ordinarie et la convergence asymptotique,” Colloqium Mathematicum, vol. 2, pp. 335–346, 1953.
6. R. C. Buck, “Generalized asymptotic density,” American Journal of Mathematics, vol. 75, pp. 335–346, 1953.
7. J. L. Kelley, General Topology, Springer, New York, NY, USA, 1955.
8. C. Kuratowski, Topology , PWN, Warszawa, India, 1958.
9. P. Kostyrko, M. Macaj, and T. Salat, “-convergence and extremal -limit points,” Mathematica Slovaca, vol. 55, pp. 443–464, 2005.
10. S. Gähler, “2-normed spaces,” Mathematische Nachrichten, vol. 28, pp. 1–43, 1964.
11. S. Gähler, “2-metrische Räumm und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963.
12. Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak, Theory of 2-Inner Product Spaces, Nova Science, Huntington, NY, USA, 2001.
13. H. Gunawan and E. K. Mashadi, “On finite dimensional 2-normed spaces,” Soochow Journal of Mathematics, vol. 27, no. 3, pp. 321–329, 2001.
14. W. Raymond, Y. Freese, and J. Cho, Geometry of Linear 2-Normed Spaces, N.Y. Nova Science, Huntington, WVa, USA, 2001.
15. M. Gürdal and S. Pehlivan, “The statistical convergence in 2-Banach spaces,” Thai Journal of Mathematics, vol. 2, no. 1, pp. 107–113, 2004.