SOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS


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1 Real Analysis Exchange Vol. 28(2), 2002/2003, pp Ruchi Das, Department of Mathematics, Faculty of Science, The M.S. University Of Baroda, Vadodara , India. Nikolaos Papanastassiou, University of Athens, Department of Mathematics, Panepistimiopolis, Athens 15784, Greece. SOME TYPES OF CONVERGENCE OF SEQUENCES OF REAL VALUED FUNCTIONS Abstract Using the notions of uniform equal and uniform discrete convergence for sequences of real valued functions, the classes of functions which are uniform equal limits and uniform discrete limits of sequences of real valued functions belonging to certain class are studied. Also, new types of convergence of sequences of real valued functions, called uniform equal, strong uniform equal and equal are defined and studied. Using uniform equal convergence, a characterization of compact metric space is obtained. 1 Introduction In recent papers [8] and [9], Papanastassiou has defined and studied the notions of uniform equal convergence and uniform discrete convergence for sequences of real valued functions. Using these convergences the author has obtained some results in measure theory. It is observed that uniform discrete convergence is stronger than uniform equal convergence as well as discrete convergence defined in [4]. On the other hand, uniform equal convergence is weaker than uniform convergence and stronger than equal convergence defined by Császár and Laczkovich in [4]. In the present paper we study the properties of classes of functions which are uniform equal limits and uniform discrete limits of sequences of functions belonging to a particular class. We also define and study Key Words: Uniform equal convergence, uniform discrete convergence, continuous convergence,  convergence, e. convergence,  s. convergence, ordinary class. Mathematical Reviews subject classification: 26A21, 26A03 Received by the editors August 8, 2001 Communicated by: Peter Bullen 43
2 44 Ruchi Das and Nikolaos Papanastassiou uniform equal convergence, strong uniform equal convergence and equal convergence which are stronger than  convergence (known as continuous convergence [10]) and obtain applications of our results in metric spaces. Section 2 contains notation and terminology used in subsequent sections. In Section 3, we study the classes Φ and Φ u.d. consisting respectively of the realvalued functions on a nonempty set X which are uniform equal limits and uniform discrete limits of sequences of functions in a particular class Φ (e.g. Theorems 3.6 and 3.8). The notion of convergence (known as continuous convergence) turned out to be useful for characterizing compactness in metric spaces ([7], Theorem 3.2, p. 129). We recall the definition of convergence. Let X be a metric space and f, f n, n N be realvalued functions defined on X. Then (f n ) converges to f (written as f n f) if for any x X and for any sequence (x n ) of points of X converging to x, (f n (x n )) converges to f(x). It is clear from the definition that this convergence is stronger than pointwise convergence. On the other hand, if the limit function f is continuous, then this convergence is weaker than uniform convergence. In Section 4, we define a new type of convergence called uniform equal convergence ( for short) which turns out to be stronger than uniform equal convergence as well as convergence. We investigate properties of this convergence and obtain a necessary and sufficient condition for a metric space to be compact in terms of it. We also define and study a notion, stronger than  convergence, called strong uniform equal convergence (s. for short). In Section 5, we introduce the notion of equal convergence which is weaker than uniform equal convergence and stronger than equal as well as  convergence. We study properties of this convergence. We do the comparative study of all these types of convergences. We end the section with some open problems concerning these convergences. 2 Notation and Terminology By N, we mean the set of all natural numbers and by R, we mean the set of all real numbers. If Γ is a set, then Γ denotes the cardinality of Γ. If x R, then [x] denotes the integer part of x. Let X be a nonempty set. By a function on X, we mean a real valued function on X. Let Φ be an arbitrary class of functions defined on X. Then we have the following definitions. Definition 2.1. A sequence of functions (f n ) in Φ is said to converge uniformly equally to a function f in Φ (written as f n f) if there exists a
3 Convergence of Sequences of Real Valued Functions 45 sequence (ε n ) n N of positive reals converging to zero and a natural number n 0 such that the cardinality of the set {n N : f n (x) f(x) ε n } is at most n 0, for each x X [8]. Definition 2.2. A sequence of functions (f n ) in Φ is said to converge uniformly discretely to a function f in Φ (written as f n f) if there exists a u.d. natural number n 0 such that the cardinality of the set {n N : f n (x) f(x) > 0} is at most n 0, for each x X (see [9]). We denote by Φ, the set of all functions on X which are uniform equal limits of sequences of functions in Φ. Similarly Φ u.d. denotes the set of all functions which are uniform discrete limits of sequences of functions in Φ. Note 2.3. One can observe that if f Φ, then for any sequence (λ n ) n N of positive reals converging to zero, there exists a sequence of functions in Φ which converges uniformly equally to f with witnessing sequence (λ n ) n N. Definition 2.4. A sequence of functions (f n ) in Φ is said to converge e. equally to f (written as f n f) if there exists a sequence (ε n ) n N of positive reals converging to zero such that, for each x X, there exists a natural number n(x) satisfying f n (x) f(x) < ε n, for each n n(x). Also, (f n ) is d. said to converge discretely to a function f in Φ (written as f n f) if, for every x X, there exists n(x) N such that f(x) = f n (x) for all n n(x) [4]. Note 2.5. For a sequence of funtions in Φ, it is clear that we have the implications: uniform convergence implies uniform equal convergence, and uniform equal convergence implies equal convergence. On the other hand uniform discrete convergence implies both discrete and uniform equal convergence. The β symbol f n f means that (fn ) does not converge to f in the respective βconvergence. Example 2.6. The following four examples show that the converse of each of the above implications fail. (i) Let f n (x) = x n for x [0, 1). Then the sequence (f n ) converges equally to the zero function on [0, 1) but not uniformly equally. (Refer to Example 4.9) (ii) Let f n be the piecewise linear function supported on [n 1, n n ] and given by x + 1 n for x [n 1, n] f n (x) = 1 for x [n, n + 1 n ] n n x for x [n + 1 n, n n ].
4 46 Ruchi Das and Nikolaos Papanastassiou Then the sequence (f n ) of continuous functions satisfies f n = 1 for each n N, and therefore does not converge uniformly to the zero function. On the other hand, it converges uniformly equally to the zero function. In fact, if (ε n ) n N is a null sequence of positive reals, then we have {n N : f n (x) ε n 3 for each x R. (iii) Let 0 < δ < 1 and f n (x) = x n for x [0, δ]. Then the sequence (f n ) converges uniformly equally to the zero function on [0, δ] but not uniformly discretely, since the set {n N : δ n > 0} is unbounded [9]. (iv) Let 0 for x (, n 1] f n (x) = x n + 1 for x [n 1, n] 1 for x [n, + ). Then the sequence (f n ) converges discretely to the zero function on R but not uniformly discretely [9]. For the function class Φ on X, we have the following definitions [5]. Definition 2.7. (a) Φ is called a lattice if Φ contains all constants and f, g Φ implies max(f, g) Φ and min(f, g) Φ. (b) Φ is called a translation lattice if it is a lattice and f Φ, c R implies f + c Φ. (c) Φ is called a congruence lattice if it is a translation lattice and f Φ f Φ. (d) Φ is called a weakly affine lattice if it is a congruence lattice and there is a set C (0, ) such that C is not bounded and f Φ, c C implies cf Φ. (e) Φ is called an affine lattice if it is a congruence lattice and f Φ, c R implies cf Φ. (f) Φ is called a subtractive lattice if it is a lattice and f, g Φ implies (f g) Φ. (g) Φ is called an ordinary class if it is a subtractive lattice, f, g Φ implies f g Φ and f Φ, f(x) 0, for all x X implies 1/f Φ.
5 Convergence of Sequences of Real Valued Functions 47 3 On the Classes of Uniform Equal and Uniform Discrete Limits We first observe the following equivalent condition for the uniform equal convergence. Theorem 3.1. Let f n, f : X R, n N. Then f n f if and only if there exists an unbounded sequence (ρ n ) n N of positive integers such that ρ n f n f 0. Proof. Suppose f n u.e f. Then there exists a sequence (ε n ) n N of positive reals converging to zero and n 0 N such that Note that {n N : f n (x) f(x) ε n } n 0, for each x X. {n N : ρ n f n (x) f(x) ε n } n 0, for each x X, ([ 1 where (ρ n ) = εn ]), is an unbounded sequence of positive integers and hence ρ n f n f 0. Conversely, if ρ n f n f 0, where (ρ n ) is an unbounded sequence of positive integers, then there exists a sequence (λ n ) of positive reals converging to zero and n 0 N such that {n N : ρ n f n (x) f(x) λ n } n 0, for each x X. ( ) λ For (θ n ) = n ρ n 0, we have {n N : f n (x) f(x) θ n } n 0, for each x X and hence f n f with witnessing sequence (θ n ). The following result describes some properties the class Φ must have if the class Φ has the properties. Theorem 3.2. Let Φ be a class of functions on X. If Φ is a lattice, a translation latice, a congruence lattice, a weakly affine lattice, an affine lattice or a subtractive lattice, then so is Φ. Proof. Suppose Φ is a lattice. Since Φ contains constant functions, Φ u.e contains constant functions. By definition it follows that if f n f, then
6 48 Ruchi Das and Nikolaos Papanastassiou f n f. Moreover, as observed in [8], if f n f, g n g and, β R then f n + βg n f + βg. Hence if f, g Φ, f n f and g n g, then ( ) fn + g n + f n g n f + g f g + = max(f, g) which implies that max(f, g) Φ. Similarly min(f, g) Φ. Thus Φ is a lattice. It is easy to observe that if Φ is a translation, a congruence, a weakly affine, an affine or a subtractive lattice, then so is Φ. We first observe the following Lemmas. Lemma 3.3. Let f n : X R, n N. If f n 0, then fn 2 0. Proof. If (λ n ) n N is witnessing sequence for uniform equal convergence of (f n ) to zero, then (λ 2 n) is witnessing sequence for uniform equal convergence of (f 2 n) to zero. Lemma 3.4. Let f n, f : X R n N. If f is bounded and f n f. Then f n f f 2. Proof. Let M be a positive real number such that f(x) M, for each x X. Since f n f, there exists a sequence (ε n ) n N of positive reals converging to zero and n 0 N such that Hence {n N : f n (x) f(x) ε n } n 0, for each x X. {n N : (f n f)(x) f 2 (x) ε n M} n 0, for each x X. Thus f n f f 2. Using the above two lemmas, we obtain the following result regarding product of uniform equal limits. Theorem 3.5. Let f, g : X R be bounded functions and f n, g n : X R, n N be such that f n f and g n g. Then f n g n f g. Proof. Since f n f and g n g, f n + g n f + g and f n g n f g. Now using Lemmas 3.3 and 3.4, we get f n g n = (f n + g n ) 2 (f n g n ) 2 4 (f + g)2 (f g) 2 = f g. 4
7 Convergence of Sequences of Real Valued Functions 49 Theorem 3.6. Let Φ be an ordinary class of functions on X. Let f Φ be bounded and such that f(x) 0 for each x X. If 1 f is bounded on X, then 1 f Φ. Proof. Let λ be such that f 2 (x) > λ > 0 for each x X. Since f Φ and f is bounded, by Theorem 3.5, f 2 Φ and hence by Note 2.3, there exists f n Φ, n N and n 0 N such that {n N : f n (x) f 2 (x) 1 n 3 } n 0, for each x X. Let g n (x) = max{f n (x), 1 n }, x X and n N. Then g n Φ for each n N. Note that {n N : g n (x) = f n (x) and g n (x) f 2 (x) 1 n 3 } n 0 and {n N : g n (x) = 1 n and g n(x) f 2 (x) 1 n 3 } n + n 0, where n = [ 1 λ] + 1. Using the fact that, {n N : g n (x) f 2 (x) 1 n 3 } ={n N : g n (x) = f n (x) and g n (x) f 2 (x) 1 n 3 } {n N : g n (x) = 1 n and g n(x) f 2 (x) 1 n 3 }, we get, {n N : g n (x) f 2 (x) 1 n 3 } n 0 + (n 0 + n ) n 1, for each x X. Therefore 1 {n N : g n (x) 1 f 2 (x) 1 n 3 n 1 λ } = {n N : g n(x) f 2 (x) g n (x) f 2 (x) 1 n 3 n 1 λ } {n N : g n (x) f 2 (x) 1 n 3 } n 1, for each x X.
8 50 Ruchi Das and Nikolaos Papanastassiou Thus f 2 Φ u.e and so f f 2 = 1 f Φ. Now, we study the properties of class Φ u.d. of uniform discrete limits of sequence of functions in a certain function class Φ. The following result follows from definition. Theorem 3.7. If Φ is a lattice, a translation lattice, a congruence lattice, a weakly affine lattice, an affine lattice or a subtractive lattice, then so is Φ u.d. We have the following result for a function class Φ which is an ordinary class. Theorem 3.8. Let Φ be an ordinary class of functions on X. Then f, g Φ u.d. implies f g Φ u.d.. Also, if f Φ u.d. is such that f(x) 0 for each x X and 1 f bounded on X then 1 f Φu.d.. Proof. Let f, g Φ u.d.. Then there exist sequences (f n ) and (g n ) in Φ such u.d. u.d. u.d. that f n f, g n g. It follows from definition that f n g n f g. Let f satisfy the assumptions. Choose λ such that f 2 (x) > λ > 0 for each x X. We first show that f 2 Φ u.d. u.d.. Let f n Φ, n N, be such that f n f. Since Φ is an ordinary class, fn 2 Φ, n N. Let (ε n ) n N be a sequence of positive reals converging to zero and g n = max{fn, 2 u.d. ε n }. Then g n Φ. Since f n f, therefore there exists n 0 N satisfying {n N : f n (x) f(x)} n 0, for each x X. Hence, {n N : g n (x) max{f 2 (x), ε n }} n 0, for each x X, which implies {n N : 1 g n (x) 1 max{f 2 (x), ε n } } n 0, for each x X. (1) Since (ε n ) n N converges to zero, there exists n N satisfying ε n < λ, for all n n. Hence, {n N : 1 max{f 2 (x), ε n } 1 f 2 (x) } < n, for each x X. (2) Now using equations (1) and (2) {n N : 1 g n(x) 1 f 2 (x) } < n 0 + n, for each x X. Hence f 2 Φ u.d., consequently f f 2 = f 1 Φ u.d.. 4 Uniform Equal Convergence The notion of convergence (known as continuous convergence) for sequences of real valued functions on a metric space turned out to be useful for characterizing compactness in metric spaces. It is known that if X is a metric space
9 Convergence of Sequences of Real Valued Functions 51 and f, f n : X R, n N are such that f n f (i.e. (fn ) converges to f), then f is continuous. Also, if X is a compact metric space, then f n f u implies f n f, where u denotes uniform convergence (see [10]). In [7], Holá and Šalát have obtained the following characterization of compact metric spaces. Theorem 4.1. A metric space (X, d) is compact if and only if for f n, f : X R, n N f n f fn u f. We define here the notion of uniform equal convergence. Definition 4.2. Let (X, d) be a metric space and f, f n : X R, n N.  Then (f n ) converges uniformly equally to f (written as f n f) if there exists a sequence (ε n ) n N of positive reals converging to zero and an n 0 N such that {n N : f n (x n ) f(x) ε n } n 0 for each x X and x n x. Remark 4.3. It is clear from this definition that  convergence implies both convergence and convergence. However, the following examples show that the converse of each of the above implications fails. Example 4.4. (i) Let f n be the characteristic function of the interval [n, n + 1 n ], n N. Then f n f 0. For if (ε n ) n N is a sequence of positive reals converging to zero then {n N : f n (x) ε n } 1, for each x R. Also f. For if x0 R then there exists n N such that for all n n f n we have x 0 < n and this implies f n (x 0 ) = 0, for all n n. If x n x 0, then given ε > 0, there exists n 0 (ε) N such that for all n n 0 (ε) we have x n (x 0 ε, x 0 + ε), but then f n (x n ) = 0, for all n max{n, n 0 (ε)}. Therefore f n (x n ) f(x 0 ) = 0. Hence f n f. Now we can observe that if { n + 1 x n = 2n if n m x 0 1 n if n > m where m N is fixed and ε n < 1 for all n N, then {n N : f n (x n ) f(x 0 ) ε n } = m. Hence (f n ) does not converges uniformly equally to the zero function. (ii) Let (f n ) be the sequence in example 2.6. (ii). Then as in the previous example, we see that (f n )  converges to the zero function but not uniformly equally.
10 52 Ruchi Das and Nikolaos Papanastassiou u.d. In the above examples, in fact f n 0. Therefore u.d.convergence need not imply convergence. Moreover, the following example shows that  convergence also need not imply u.d.convergence. Example 4.5. Let f n : R R be defined by f n (x) = 1 n, n N and f 0 on  R. Then note that f n f but f n u.d. f. Remark 4.6. (i) Let (X, d) be a metric space and f n, f : X R, n = 1, 2,...  such that f n f. Then f n f and hence f is continuous even if the fn are not (see [10]). Thus  convergence implies that the limit function is continuous. (ii) In general, uniform convergence need not imply uniform equal convergence. For example if f is a discontinuous function from X to R and f n = f, u for all n N, then f n f but since f is discontinuous, fn does not converge uniformly equally to f. However, we have the following result. Theorem 4.7. Let X be a metric space and f n : X R, n N. If the sequence (f n ) converges uniformly to the zero function, then the sequence (f n ) converges uniformly equally to the zero function. u Proof. Since f n 0, there exists a sequence (εn ) n N of positive reals converging to zero and n 0 N such that f n (x) < ε n, for all n n 0 and for each x X. This gives {n N : f n (x n ) ε n } n 0 for every converging  sequence (x n ) in X. Hence f n 0. In the converse direction, we have the following result. Theorem 4.8. Let (X, d) be a compact metric space and f n, f : X R,  u n N. Then f n f f n f. Proof. It follows from the fact that f n  f f n f fn u f, as X is a compact metric space (see [10]). The following example shows that convergence need not imply uniform equal convergence. Example 4.9. Let f n : (0, 1) R be defined by f n (x) = x n, n N and f 0. Then f n f. Let 0 < δ < 1 and xn (0, 1) be such that x n δ. If δ < ϑ < 1, then there exists n 0 N such that for all n n 0 we have x n < ϑ. But then f n (x n ) = x n n < ϑ n. So f n (x n ) 0 = f(δ), since ϑ n 0. Hence f n 0. However, fn f. For if (ε n ) n N is a sequence of positive reals converging to zero and 0 < ε < 1, then there exists n 0 N such that for all n n 0, ε n < ε. Consequently we have {n N : x n ε} [n 0, + ) {n N : x n ε n } [n 0, + ).
11 Convergence of Sequences of Real Valued Functions 53 But the function n ε (x) = {n N : x n ε}, x (0, 1) is unbounded and therefore the function n(x) = {n N : x n ε n }, x (0, 1) is unbounded. Hence f n f (see also [9]). Note (i) The previous example also shows that convergence need not imply uniform equal convergence. In addition we have that the sequence (f n ) converges equally to 0 on (0, 1), since (0, 1) = k=2 [ 1 k, 1 1 k ] and (f n) converges uniformly to 0 on [ 1 k, 1 1 k ] for every k N {1}. So we have here a simple example which distinguishes convergence from uniformly equally convergence, and at the same time the equal convergence from uniformly equal convergence. (ii) Examples in Remark 4.6 (ii) shows that uniform equal convergence need not imply convergence (since f being discontinuous, f n f). Also, the following example shows the same. Example Let 1 n for x [0, 1 n ] f n (x) = 0 for x ( 1 n, 1) 1 for x = 1. n N and f : [0, 1] [0, 1] be defined by { 1 for x = 1 f(x) = 0 otherwise. It is easy to verify that f n f. f n u f and hence fn f but f being discontinuous We now obtain a characterization of compact metric spaces using uniform equal convergence. Theorem A metric space (X, d) is compact if and only if the convergence of a sequence (f n ) of real valued functions defined on X to the zero function implies the uniform equal convergence of the sequence (f n ) to the zero function. Proof. If X is compact metric space, then by Theorem 4.1, f n 0 u a f n 0 and hence by Theorem 4.7 fn 0. Conversely, suppose (X, d) is not a compact metric space. We first recall the construction of maps fp in Theorem 3.1 in [7]. Since X is not compact, there exists a sequence (x k )
12 54 Ruchi Das and Nikolaos Papanastassiou of distinct points of X such that there exists no convergent subsequence of (x k ). Since every point of the set {x 1,... x n,...} is an isolated point of the set {x 1, x 2,..., x n,...}, there exist δ k > 0, k = 1, 2,... such that δ k 0 as k and the closed balls B[x k, δ k ] = {x X d(x, x k ) δ k }, k = 1, 2,... are pairwise disjoint. Then H = k=1 B[x k, δ k ] is a closed set. Define a sequence (f p ) of real valued functions on the set {x 1, x 2,..., x n,...} by f 1 (x n ) = 0, n = 1, 2,... and for p > 1, f p (x m ) = (1 1/m) p 1 if 1 m p and f p (x p+j ) = f p (x p ) for j = 1, 2,.... Define for p N, fp (x) = 0 if x H and fp (x) = f p (x j ) (δ j d(x, x j ))/δ j, if x B[x j, δ j ] (j = 1, 2,...). Then as proved in [7], fp 0. However, the fact that fp (x p ) = (1 1 p )p 1 is decreasing and converges to e 1, where e is Euler number, implies that (fp ) does not converge uniformly equally to the zero function. For if (ε n ) n N is a null sequence of positive reals, then there exists n 0 N such that, for all n n 0 we have ε n < e 1. Let ε (ε n0, e 1 ). Then for all p N, Also we have, {n N : f n(x p ) ε} p 2. {n N, n n 0 : f n(x p ) ε} {n N, n n 0 : f n(x p ) ε n }. Therefore for all p N {n N, n n 0 : f n(x p ) ε n } p 2 n 0. Hence fp 0. Now, since  convergence implies convergence, we have that the sequence (fp ) does not converges uniformly equally to the zero function. The following notion was introduced by A. Denjoy [6]. The series n=0 f n of realvalued function converges pseudonormally on a set X, if and only if there exists a convergent series n=0 ε n of positive reals such that for every x X there exists an index k x such that f k (x) < ε k for every k > k x. In [3], the authors define a sequence (f n ), of real valued functions, to converge pseudonormally to a real valued function f if there exists a sequence (ε n ) n N of positive reals with n=1 ε n < + such that for each x X, there exists n 0 (x) N satisfying f n (x) f(x) ε n for all n n 0 (x). The secondnamed author of the present paper introduced in [8] a stronger notion of convergence, called strong uniform equal, which is defined as follows. Let f n, f : X R, n = 1, 2,.... Then (f n ) converges to f strongly s. uniformly equally (written as f n f), if there exists a sequence (ε n ) n N of positive reals with 1 ε n < and n 0 N such that {n N : f n (x) f(x) ε n } n 0, for each x X.
13 Convergence of Sequences of Real Valued Functions 55 From the definition it follows that strong uniform equal convergence is stronger than uniform equal convergence as well as pseudonormal convergence. However, the following examples show that they are not the same. Example (i) Let f n : R R, n N be defined by f n (x) = 1 n, x R. s. Then f n 0 but f n 0. (ii) Let f n be the characteristic function of the interval [n, ), n N. Then (f n ) converges pseudonormally to f 0 on R but not strongly uniformly equally. Here we define the notion of strong uniform equal convergence. Definition We say that a sequence of real valued functions (f n ) converges strongly uniformly equally to a function f (written as f n s. f), if there exists a convergent series n=0 ε n of positive reals and an index n 0 N such that {n N : f n (x n ) f(x) ε n } n 0, for every x X and x n x. Remark It is clear that the notion of strong uniform equal convergence is stronger than uniform equal convergence and strong uniform equal convergence both. f n The following examples distinguish these types of convergence.  Example (i) Let f n (x) = 1 n, x R. Then f n f 0 but s. f 0 (ii) Let f n be the characteristic function on [n, n + 1 n ], n N and f 0 on R. Then f n s. 0 but f n s. 0. We recall now the definition of strong uniform convergence defined in [8]. Let f, f n : X R, n N. Then (f n ) is said to converge strongly s.u. uniformly to f (written as f n f) on X, if and only if, there exists a sequence of positive reals (ε n ) with n=1 ε n < + and an index n 0 N such that f n (x) f(x) < ε n, for all n n 0 and for every x X. Note that Example 4.16 (i) shows that s.u. convergence is stronger than uniform convergence. It is easy to observe the following result. Theorem Let f n : X R, n = 1, 2,... and f : X R be the zero s.u. s. function such that f n f. Then f n f.
14 56 Ruchi Das and Nikolaos Papanastassiou 5 Equal Convergence We define the following convergence which is intermediate between  convergence and convergence. Definition 5.1. Let f, f n, n N be functions on X. We say that the sequence e. (f n ) converges equally to f (written as f n f) if there exists a sequence (ε n ) n N of positive reals converging to zero such that for each x X and sequence (x n ) of points of X such that x n x, there exists a natural number n 0 n 0 (x, (x n )) satisfying f n (x n ) f(x) < ε n for all n n 0. Remark 5.2. It follows from the definition that: (i) equal convergence implies equal convergence. (ii) If f n e. f, then f n f. (iii) If f n  f, then f n e. f. (iv) If f n e. f, then f is continuous. Example 5.3. We give some examples which distinguish all these types of convergence. (i) Let X = [0, 1] and f n (x) = x n, n N and f be defined by f(x) = 0, if e x [0, 1), f(x) = 1 if x = 1. Then f n f but (fn ) does not converge equally to f since f is not continuous. (ii) In examples 4.4, f n e. f but f n  f. Note 5.4. The following figure shows the relation between all the convergences discussed here. The numbers at the arrows refer to the examples showing that those implications are not true.
15 Convergence of Sequences of Real Valued Functions (iv) d. e (i) (iii) u.d.   e (f constant) s. 4.6 (ii) 4.16 (i) 7 u a  s (ii) Note 5.5. We recall that a convergence structure is said to be an Lspace if every subsequence of a convergent sequence converges to the same limit and if every constant sequence converges to its common value (compare [1]). According to Remark 1.2 (ii) in [8], the space R X of all real valued functions defined on X with uniform equal convergence is an Lspace. Also, from the definitions, it follows that the space R X with each of uniform discrete convergence, uniform equal convergence, equal convergence and convergence is an Lspace. Open Problems (i) Let X be a topological space. We call X an space if p.w. whenever f n : X R, n N, continuous, are such that f n 0 then f n 0 p.w. and we call X a weak space if whenever f n 0, where p.w. denotes pointwise convergence, then there exists a subsequence say, (f nk ) of (f n ) such that f nk 0. Such kind of study has been done in [2]. Clearly an space is a weak space and R with cocountable topology τ c is an space. (For if p.w. f n 0 and x n x 0, x 0 X, then there exists n 0 N such that for all n n 0 x n = x 0, and therefore f n (x n ) = f n (x 0 ) 0 = f(x 0 ).) It would be p.w. interesting to study spaces, and in a similar way, spaces in which f n 0 e. p.w.  implies f n 0 or f n 0 implies f n 0. e (ii) Characterize the topological spaces in which f n 0 implies fn 0, where f n, n N are continuous. In a similar way, characterize spaces in which  f n 0 implies fn 0. We recall that compact metric spaces are examples  of such spaces, as on a compact metric space f n 0 implies fn 0. (iii) Find a characterization of convergence, e. and  convergence analogous to the characterization of strong equal convergence obtained in [8].
16 58 Ruchi Das and Nikolaos Papanastassiou (iv) Finally, (in the usual notations) find the conditions under which (a) Φ  = Φ, (b) Φ  = Φ, (c) Φ  = Φ s., (d) Φ u.d. = Φ, (e) Φ = Φ e. Acknowledgment. This work was done when the firstnamed author was visiting the Department of Mathematics, University of Athens, Greece, under the Greek Government Scholarship (September June 2001). The author extends her sincere thanks to the department for providing excellent hospitality during her stay. The authors thank the referee for his valuable suggestions. References [1] Z. Bukovská, L. Bukovský and J. Ewert, Quasiuniform convergence and Lspaces, Real Anal. Exchange, 18 (1992/93), [2] L. Bukovský, I. Reclaw and M. Repický, Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology and its Appl., 41 (1991), [3] L. Bukovský, I. Reclaw and M. Repický, Spaces not distinguishing convergences of realvalued functions, Topology and its Appl., 112 (2001), [4] Á. Császár and M. Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar., 10 (1975), [5] Á. Császár and M. Laczkovich, Some remarks on discrete Baire classes, Acta Math. Acad. Sci. Hungar., 33 (1979), [6] A. Denjoy, Lecons sur le calcul des coefficients d une série trigonométrique, 2 e Partie, Paris, [7] Ľ. Holá and T. Šalát, Graph convergence, uniform, quasiuniform and continuous convergence and some characterizations of compactness, Acta Math. Univ. Comenian, (1988),
17 Convergence of Sequences of Real Valued Functions 59 [8] N. Papanastassiou, On a new type of convergence of sequences of functions, submitted. [9] N. Papanastassiou, Modes of convergence of real valued functions. Preprint. [10] S. Stoilov, Continuous convergence, Rev. Math. Pures Appl., 4 (1959),
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