International Journal of Analysis and Applications
ISSN 2291-8639
Volume 15, Number 2 (2017), 229-237
DOI: 10.28924/2291-8639-15-2017-229

QUASI-SLOWLY OSCILLATING SEQUENCES IN LOCALLY NORMAL RIESZ

SPACES

BIPAN HAZARIKA1 AND AYHAN ESI2,∗

Abstract. In this paper, we study quasi-slowly oscillating sequences, on quasi-slowly oscillating

compactness and quasi-slowly oscillating continuous functions in locally normal Riesz space.

1. Introduction

The concept of continuity and any concept involving continuity play a very important role not
only in pure mathematics but also in other branches of sciences involving mathematics especially in
computer science, information theory, biological science, speech analysis, bioinformatics.

A real valued function is continuous on the set of real numbers if and only if it preserves Cauchy
sequences. Using the idea of continuity of a real function and the idea of compactness in terms of
sequences, many kinds of continuities were introduced and investigated, not all but some of them we
recall in the following: forward continuity [6], slowly oscillating continuity [9, 12, 14, 15, 35], statistical
ward continuity [7], δ-ward continuity [11], ideal ward continuity [5, 22]. The concept of a Cauchy
sequence involves far more than that the distance between successive terms is tending to zero. Nev-
ertheless, sequences which satisfy this weaker property are interesting in their own right. A sequence
(xn) of points in R is called quasi-Cauchy if (∆xn) is a null sequence where ∆xn = xn+1 −xn. In [4]
Burton and Coleman named these sequences as ”quasi-Cauchy” and in [8] Çakallı used the term ”ward
convergent to 0” sequences. In terms of quasi-Cauchy we restate the definitions of ward compactness
and ward continuity as follows: a function f is ward continuous if it preserves quasi-Cauchy sequences,
i.e. (f(xn)) is quasi-Cauchy whenever (xn) is, and a subset E of R is ward compact if any sequence
x = (xn) of points in E has a quasi-Cauchy subsequence z = (zk) = (xnk) of the sequence x.

A Riesz space is an ordered vector space which is a lattice at the same time. It was first introduced
by F. Riesz [32] in 1928. Riesz spaces have many applications in measure theory, operator theory and
optimization. They have also some applications in economics (see [2]), and we refer to [1, 3, 21, 23, 25,
27, 28, 30, 31, 33, 36] for more details.

2. Preliminaries and Notations

It is known that a sequence x = (xn) of points in R, the set of real numbers, is slowly oscillating,
denoted by x ∈ so, if

lim
λ→1+

limn max
n+1≤k≤[λn]

|xk −xn| = 0

where [λn] denotes the integer part of λn. This is equivalent to the following if (xm−xn) → 0 whenever
1 ≤ m

n
→ 1 as m,n →∞. Using ε > 0 and δ this is also equivalent to the case when for any given ε > 0,

there exists δ = δ(ε) > 0 and N = N(ε) such that |xm −xn| < ε if n ≥ N(ε) and n ≤ m ≤ (1 + δ)n
(see [9]).

A function defined on a subset E of R is called slowly oscillating continuous if it preserves slowly
oscillating sequences, i.e. (f(xn)) is slowly oscillating whenever (xn) is.

Connor and Grosse-Erdman [13] gave sequential definitions of continuity for real functions calling
G-continuity instead of A-continuity and their results covers the earlier works related to A-continuity

Received 2nd July, 2017; accepted 26th August, 2017; published 1st November, 2017.
2010 Mathematics Subject Classification. 46A19; 40A05; 40G15; 46A50; 54E35.
Key words and phrases. Riesz space; continuity; quasi-Cauchy sequence; quasi-slowly oscillating sequences.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

229



230 HAZARIKA AND ESI

where a method of sequential convergence, or briefly a method, is a linear function G defined on a
linear subspace of s, space of all sequences, denoted by cG, into R. A sequence x = (xn) is said to be
G-convergent to ` if x ∈ cG and G(x) = `. In particular, lim denotes the limit function lim x = limn xn
on the linear space c.

A method G is called regular if every convergent sequence x = (xn) is G-convergent with G(x) =
lim x. A method is called subsequential if whenever x is G-convergent with G(x) = `, then there is a
subsequence (xnk) of x with limk xnk = ` (for details see [10]).

Let X be a real vector space and ≤ be a partial order on this space. Then X is said to be an ordered
vector space if it satisfies the following properties:

(i) if x,y ∈ X and y ≤ x, then y + z ≤ x + z for each z ∈ X.
(ii) if x,y ∈ X and y ≤ x, then ay ≤ ax for each a ≥ 0.
If, in addition, X is a lattice with respect to the partial ordered, then X is said to be a Riesz space

(or a vector lattice)(see [36]).
For an element x of a Riesz space X, the positive part of x is defined by x+ = x∨ 0 = sup{x, 0},

the negative part of x by x− = −x∨ 0 and the absolute value of x by |x| = x∨ (−x), where 0 is the
zero element of X.

A subset S of a Riesz space X is said to be normal if y ∈ S and |x| ≤ |y| implies x ∈ S.
A topological vector space (X,τ) is a vector space X which has a topology (linear) τ, such that the

algebraic operations of addition and scalar multiplication in X are continuous. Continuity of addition
means that the function f : X × X → X defined by f(x,y) = x + y is continuous on X × X, and
continuity of scalar multiplication means that the function f : R ×X → X defined by f(a,x) = ax is
continuous on R×X.

Every linear topology τ on a vector space X has a base N for the neighborhoods of θ satisfying the
following properties:

(1) Each Y ∈ N is a balanced set, that is, ax ∈ Y holds for all x ∈ Y and for every a ∈ R with
|a| ≤ 1.

(2) Each Y ∈ N is an absorbing set , that is , for every x ∈ X, there exists a > 0 such that ax ∈ Y.
(3) For each Y ∈ N there exists some E ∈ N with E + E ⊆ Y.
A linear topology τ on a Riesz space X is said to be locally normal or solid if τ has a base at zero

consisting of normal sets. A locally normal (solid) Riesz space (X,τ) is a Riesz space equipped with
a locally normal (solid) topology τ.

Recall that a first countable space is a topological space satisfying the ”first axiom of countability”.
Specifically, a space X is said to be first countable if each point has a countable neighborhood basis
(local base). That is, for each point x in X there exists a sequence V1,V2, · · · of open neighborhoods
of x such that for any open neighborhood V of x there exists an integer j with Vj contained in V.

The idea of statistical convergence first appeared, under the name of almost convergence, in the first
edition Zygmund [37] of celebrated monograph [38] of Zygmund. Later, this idea was introduced by
Fast [16] and Steinhaus [34] and many authors. Actually, this concept is based on the natural density
of subsets of N of positive integers. A subset E of N is said to have natural or asymptotic density
δ(E), if

δ(E) = lim
n→∞

|E(n)|
n

exists,

where E(n) = {k ≤ n : k ∈ E} and |E| denotes the cardinality of the set E.
A sequence x = (xn) of points in X is said to be statistically convergent (see [1]) to an element L

in X if for each τ- neighborhood V of zero, δ({n ∈ N : xn −L /∈ V}) = 0, i.e.

lim
k→∞

1

k
|{n ≤ k : xn −L /∈ V}| = 0.

Kostyrko et al. [26] introduced the notion of ideal convergence which is a generalization of statistical
convergence (see [16,18]) based on the structure of the admissible ideal I of subsets of natural numbers
N.

A family of sets I ⊂ P(N) (the power sets of N) is said to be an ideal on N if and only if φ ∈ I
for each A,B ∈ I, we have A∪B ∈ I for each A ∈ I and each B ⊂ A, we have B ∈ I. A non-empty
family of sets F ⊂ P(N) is said to be a filter on N if and only if φ /∈ F for each A,B ∈ F, we have
A∩B ∈ F each A ∈ F and each B ⊃ A, we have B ∈ F. An ideal I is called non-trivial ideal if I 6= φ



QUASI-SLOWLY OSCILLATING SEQUENCES IN LOCALLY NORMAL RIESZ SPACES 231

and N /∈ I. Clearly I ⊂ P(N) is a non-trivial ideal if and only if F = F(I) = {N−A : A ∈ I} is a filter
on N. A non-trivial ideal I ⊂ P(N) is called admissible if and only if {{n} : n ∈ N} ⊂ I. Throughout
we assume I is a non-trivial admissible ideal in N.

A sequence x = (xn) of points in a locally normal Riesz space X is said to be ideally convergent to
x0 ∈ X if for every τ-neighborhood V of zero, the set {n ∈ N : xn−x0 /∈ V}∈ I. In this case we write
xn

Iτ→ ` i.e. Iτ -lim xn = ` (for details see [21]).
The notion of lacunary statistical convergence was introduced by Fridy and Orhan [19] and has

been investigated for the real case in [20]. A sequence x = (xn) in X is called lacunary statistically
convergent to an element L in X (see [29]) if for every τ-neighborhood V of zero,

lim
r→∞

1

hr
|{n ∈ Jr : xn −L /∈ V}| = 0

where J = (kr−1,kr] and k0 = 0,hr := kr − kr−1 → ∞ as r → ∞ and (θ) = (kr) is an increasing
sequence of positive integers.

Throughout the article, the symbol Nnor we will denote any base at zero consisting of normal sets
and satisfying the conditions (1), (2) and (3) in a locally normal topology. Also (X,τ) a locally normal
Riesz space (in short LNRS) and N and R will denote the set of all positive integers, and the set of all
real numbers, respectively. We will use boldface letters x, y, z, . . . for sequences x = (xn), y = (yn),
z = (zn), . . . of points in X.

3. Quasi-slowly oscillating sequences in LNRS

In this section we introduce the concepts of quasi-slowly oscillating continuity and quasi-slowly
oscillating compactness in LNRS and establish some interesting results related to these notions.

A sequence x = (xn) of points in X is called quasi-Cauchy if for each τ-neighborhood V of zero,
there exists an m0 ∈ N such that xn+1 − xn ∈ V for n ≥ m0. It is clear that Cauchy sequences are
slowly oscillating not only the real case but also in the LNRS setting. It is easy to see that any slowly
oscillating sequence of points in X is quasi-Cauchy and therefore Cauchy sequence is quasi-Cauchy.
The converses are not always true. There are quasi-Cauchy sequences which are not Cauchy. There
are quasi-Cauchy sequences which are not slowly oscillating. Any subsequence of Cauchy sequence is
Cauchy. The analogous property fails for quasi-Cauchy sequences and slowly oscillating sequences as
well. A sequence x = (xn) of points in X is said to be slowly oscillating, denoted by x ∈ so(X) if (xn)
is a slowly oscillating sequences, i.e. for each τ-neighborhood V of zero, there exist δ = δ(V ) > 0 and
m = m(V ) such that

xk −xn ∈ V for n ≥ m(V ) and n ≤ k ≤ (1 + δ)n.
A sequence x = (xn) of points in X is called ideally quasi-Cauchy if for each τ-neighborhood V of

zero, the set

{n ∈ N : xn+1 −xn /∈ V}∈ I.
It is clear that slowly oscillating sequence of points in X is ideally quasi-Cauchy.

Now we introduce the notion of quasi-slowly oscillating sequences and quasi-slowly oscillating con-
tinuity in LNRS.

Definition 3.1. A sequence x = (xn) of points in X is said to be quasi-slowly oscillating, denoted by
x ∈ qso(X) if (∆xn) is a slowly oscillating sequences, i.e. for each τ-neighborhood V of zero, there
exist δ = δ(V ) > 0 and m = m(V ) such that

∆xk − ∆xn ∈ V for n ≥ m(V ) and n ≤ k ≤ (1 + δ)n.

It is clear that a convergent sequence is slowly oscillating, since every convergent sequence is a
Cauchy sequence, and any slowly oscillating sequence is quasi-Cauchy, but the converse need not to
be true in general. For examples, if X = R, then (

∑∞
n=1

1
n

), (ln n), (ln ln n) are slowly oscillating, but

not Cauchy. The sequence (
∑k
n=1

1
n

) is quasi-Cauchy, but not slowly oscillating.

Theorem 3.1. If a sequence is slowly oscillating then it is a quasi-slowly oscillating.



232 HAZARIKA AND ESI

Proof. Let (xn) be a slowly oscillating sequence. For each τ-neighborhood V of zero, there exists a
Y ∈ Nnor such that Y ⊆ V. Choose W ∈ Nnor such that W −W ⊆ Y. Since (xn) is a slowly oscillating
sequence, there exist δ = δ(W) > 0 and a positive integer n1 = n1(W) such that

xk −xn ∈ W for all n ≥ n1 and n ≤ k ≤ (1 + δ)n.

Hence for all n ≥ n1(W) and n ≤ k ≤ (1 + δ)n we have

∆xk − ∆xn = (xk −xk+1) − (xn −xn+1) = (xk −xn) − (xk+1 −xn+1) ∈ W −W ⊆ Y ⊆ V.

It implies that (xn) is a quasi-slowly oscillating sequence. �

Definition 3.2. A function f defined on a subset E of X is called quasi-slowly oscillating continuous
if it transforms quasi-slowly oscillating sequences to quasi-slowly oscillating sequences of points in E,
that is, (f(xn)) is quasi-slowly oscillating whenever (xn) is quasi-slowly oscillating sequences of points
in E.

We note that sum of two quasi-slowly oscillating continuous functions is quasi-slowly oscillating
continuous and the composite of two quasi-slowly oscillating continuous functions is quasi-slowly os-
cillating continuous in LNRS.

In connection with slowly oscillating sequences, quasi-slowly oscillating sequences and convergent
sequences the problem arises to investigate the following types of continuity of functions on X.

(qso-qso): : (xn) ∈ qso(X) ⇒ (f(xn)) ∈ qso(X)
(qso-c): : (xn) ∈ qso(X) ⇒ (f(xn)) ∈ c(X)

(c-c): : (xn) ∈ c(X) ⇒ (f(xn)) ∈ c(X)
(c-qso): : (xn) ∈ c(X) ⇒ (f(xn)) ∈ qso(X)

(qso-so): : (xn) ∈ qso(X) ⇒ (f(xn)) ∈ so(X)
(so-qso): : (xn) ∈ so(X) ⇒ (f(xn)) ∈ qso(X)
(u): : uniform continuity of f.

It is clear that (qso-qso) implies (so-qso), but (so-qso) need not imply (qso-qso). Also (qso-c)
implies (c-qso) and (qso-c) implies (c-c) and we see that (c-c) need not imply (qso-c), because
identity function is an example for it. We also see that (u) implies (so-qso).

Theorem 3.2. If f is quasi-slowly oscillating continuous on a subset E of X then it is continuous on
E in the ordinary sense.

Proof. Suppose that f is quasi-slowly oscillating continuous on E and let (xn) be any convergent
sequence of points in E with limn→∞xn = x0. Then the sequence

(x1,x1,x0,x0,x2,x2,x0,x0, ...,xn−1,xn−1,x0,x0,xn,xn,x0,x0, ...)

is also convergent to x0 and hence (yn) is quasi-slowly oscillating. Since f is quasi-slowly oscillating
continuous, the sequence

(f(x1),f(x1),f(x0),f(x0),f(x2),f(x2),f(x0),f(x0), ...,

f(xn−1),f(xn−1),f(x0),f(x0),f(xn),f(xn),f(x0),f(x0), ...)

is also quasi-slowly oscillating. From the definition of quasi-slowly oscillation, we have that the sequence

(0,f(x1) −f(x0), 0,f(x0) −f(x2), 0,f(x2) −f(x0), 0,f(x0) −f(x3), 0,f(x3) −f(x0), 0, ...,

0,f(x0) −f(xn−1), 0,f(xn−1) −f(x0),f(x0) −f(xn), 0,f(xn) −f(x0), ...)
is a slowly oscillating. Since any slowly oscillating sequence is quasi-Cauchy, then the sequence

(f(x0) −f(x1),f(x1) −f(x0),f(x0) −f(x2),f(x2) −f(x0),f(x0) −f(x3),f(x3) −f(x0), ...,

f(x0) −f(xn−1),f(xn−1) −f(x0),f(x0) −f(xn),f(xn) −f(x0), ...)
is a null sequence. Now it follows that if for each τ-neighborhood V of zero, there exists m = m(V )
such that

f(xn) −f(x0) ∈ V for n ≥ m.
This completes the proof of theorem. �



QUASI-SLOWLY OSCILLATING SEQUENCES IN LOCALLY NORMAL RIESZ SPACES 233

In general the converse is not true. If X = R. Then, it follows from the function f(x) = x2 + 1 and
the sequence (xn) = (

√
n).

Corollary 3.1. Any quasi-slowly oscillating continuous function is G-continuous for any regular sub-
sequential method G.

Corollary 3.2. If f is quasi-slowly oscillating continuous on a subset E of X, then it is ideally
continuous on E.

Corollary 3.3. If f is quasi-slowly oscillating continuous on a subset E of X, then it is statistically
continuous on E.

Corollary 3.4. If f is quasi-slowly oscillating continuous on a subset E of X, then it is lacunary
statistically continuous on E.

Theorem 3.3. If f is a uniformly continuous function defined on a subset E of X, then it is quasi-
slowly oscillating continuous on E.

Proof. Let f be uniformly continuous function and x = (xn) be any quasi-slowly oscillating sequence
in E. Let W be a τ-neighborhood of zero. Since f is uniformly continuous on E, then there exists
a τ-neighborhood V of zero such that f(x) − f(y) ∈ W whenever x − y ∈ V. Since (xn) is slowly
oscillating, for the same τ-neighborhood W of zero, there exist m = m(V ) and δ = δ(V ) > 0 such that
∆xk −∆xn ∈ V for n ≥ m(V ) and n ≤ k ≤ (1 + δ)n. Hence we have ∆f(xk)−∆f(xn) ∈ W whenever
n ≥ m(V ) and n ≤ k ≤ (1 + δ)n. It follows that (f(xn)) is quasi-slowly oscilatting. This completes
the proof of theorem. �

Definition 3.3. A sequence (xn) of points in X is called Cesáro quasi-slowly oscillating if (tn) is
-quasi-slowly oscillating, where tn =

1
n

∑n
k=1 xk, is the Cesáro means (see [17]) of the sequence (xn).

Also a function f defined on a subset E of X is called Cesáro quasi-slowly oscillating continuous if it
preserves Cesáro quasi-slowly oscillating sequences of points in E.

By using the similar argument used in proof of Theorem 3.3, we immediately have the following
result.

Theorem 3.4. If f is a uniformly continuous on a subset E of X and (xn) is a quasi-slowly oscillating
sequence in E, then (f(xn)) is Cesáro quasi-slowly oscillating.

Definition 3.4. A sequence of functions (fn) defined on a subset E of X is said to be uniformly
convergent to a function f if for each τ-neighborhood V of zero, there exists an integer n0 = n0(V )
such that fn(x) −f(x) ∈ V for all n ≥ n0 and x ∈ E.

Theorem 3.5. If (fn) is a sequence of quasi-slowly oscillating continuous functions defined on a
subset E of X and (fn) is uniformly convergent to a function f on E, then f is quasi-slowly oscillating
continuous on E.

Proof. Let (xn) be any quasi-slowly oscillating sequence of points in E. By uniform convergence of
(fn), if for each τ-neighborhood V of zero, there exists a Y ∈ Nnor such that Y ⊆ V. Choose W ∈ Nnor
such that −W + W + W −W + W ⊆ Y. Then there exists n1 = n1(W) such that

fn(x) −f(x) ∈ W
for each x ∈ E and for all n ≥ n1(W). Also since fn1 is quasi-slowly oscillating continuous, there exist
n2 = n2 > n1 and δ = δ(W) > 0 such that

∆fn1 (xk) − ∆fn1 (xn) ∈ W
whenever n ≥ n2(W) and n ≤ k ≤ (1 + δ)n. Therefore if n ≥ n1(W) and n ≤ k ≤ (1 + δ)n we have

∆f(xk) − ∆f(xn) = [f(xk) −f(xk+1)] − [f(xn) −f(xn+1)]
= [f(xk) −fn1 (xk)] + [fn1 (xk+1) −f(xk+1)] + [fn1 (xn) −f(xn)] + [f(xn+1) −fn1 (xn+1)]

+[fn1 (xk) −fn1 (xk+1) −fn1 (xn) + fn1 (xn+1)] ∈−W + W + W −W + W ⊆ Y ⊆ V.
Thus it implies that ∆f(xk) − ∆f(xn) ∈ V if n ≥ n1 and n ≤ k ≤ (1 + δ)n. It follows that (f(xn)) is
a quasi-slowly oscillating sequences of points in E which completes the proof of theorem. �



234 HAZARIKA AND ESI

Using the same techniques as in the Theorem 3.5, the following result can be obtained easily.

Theorem 3.6. If (fn) is a sequence of Cesáro quasi-slowly oscillating continuous functions defined on
a subset E of X and (fn) is uniformly convergent to a function f on E, then f is Cesáro quasi-slowly
oscillating continuous on E.

Theorem 3.7. The set of all quasi-slowly oscillating continuous functions defined on a subset E of
X is a closed subset of all continuous functions on E, that is qso(E) = qso(E), where qso(E) is the

set of all quasi-slowly oscillating continuous functions defined on E and qso(E) denotes the set of all
cluster points of qso(E).

Proof. Let f be any element of qso(E). Then there exists a sequence of points (fn) in qso(E) such
that limn→∞fn = f. To show that f is quasi-slowly oscillating sequence on E. Now let (xn) be any
quasi-slowly oscillating sequence in E. Let V be an arbitrary τ-neighborhood of zero. There exists a
Y ∈ Nnor such that Y ⊆ V. Choose W ∈ Nnor such that W +W +W +W +W ⊆ Y. Since (fn) converges
to f, there exists a positive integer n1 such that for all x ∈ E and for all n ≥ n1, fn(x) −f(x) ∈ W.
Also since fn1 is quasi-slowly oscillating continuous, there exist an integer n2 = n2 > n1 and δ > 0
such that

∆fn1 (xk) − ∆fn1 (xn) ∈ W
whenever n ≥ n2 and n ≤ k ≤ (1 + δ)n. Hence, for all n ≥ n1 and n ≤ k ≤ (1 + δ)n we have

∆f(xk) − ∆f(xn) = [∆f(xk) − ∆fn1 (xn)] + [∆fn1 (xn) − ∆fn1 (xk)] + [∆fn1 (xk) − ∆f(xn)]
= [f(xk) −f(xk+1)] − [fn1 (xn) −fn1 (xn+1)] + [fn1 (xn) −fn1 (xn+1)] − [fn1 (xk) −fn1 (xk+1)]

+[fn1 (xk) −fn1 (xk+1)] − [f(xn) −f(xn+1)]
∈ W + W + W + W + W ⊆ Y ⊆ V.

Thus it implies that f(xk)−f(xn) ∈ V for all n ≥ n1 and n ≤ k ≤ (1 +δ)n. Thus f is slowly oscillating
continuous function on E and this completes the proof of theorem. �

Corollary 3.5. The set of all quasi-slowly oscillating continuous functions defined on a subset E of
X is a complete subspace of the space of all continuous functions on E.

An element x0 in X is called an ideal limit point of a subset E of X if there is an E-valued sequence
of points with ideal limit x0. It follows that the set of all ideal limit points of E is equal to the set of
all limit points of E in the ordinary sense. An element x0 in X is called an ideal accumulation point
of a subset E if it is an ideal limit point of the set E −{x0}. The set of all ideal accumulation points
of E is equal to the set of all accumulation points of E in the ordinary sense.

A function f on X is said to have an ideally sequential limit at a point x0 of X if the image sequence
(f(xn)) is ideally convergent to x0 for any ideally convergent sequence x = (xn) with ideal limit x0
and a function f is to be ideally sequentially continuous at a point x0 of X if the sequence (f(xn))
is ideally convergent to f(x0) for any ideally convergent sequence x = (xn) with ideal limit x0 (for
details see [5]).

Lemma 3.1. A function f on X has an ideally sequential limit at a point x0 of X if and only if it
has an ideal limit at a point x0 of X in ordinary sense.

Proof. The proof follows from the fact that any ideally convergent sequence has a convergent subse-
quence (also see [5]). �

Next we define the concept of quasi-slowly oscillating compactness in LNRS.

Definition 3.5. A subset E of X is called quasi-slowly oscillating compact if any sequence of points
in E has a quasi-slowly oscillating subsequence.

We see that any compact subset of X is quasi-slowly oscilatting compact, union of two quasi-
slowly oscillating compact subsets of X is quasi-slowly oscillating compact. Any subset of quasi-slowly
oscillating compact set is also quasi-slowly oscillating compact and so intersection of any quasi-slowly
oscillating compact subsets of X is quasi-slowly oscillating compact.

Theorem 3.8. A quasi-slowly oscillating continuous image of a quasi-slowly oscillating compact subset
of X is quasi-slowly oscillating compact.



QUASI-SLOWLY OSCILLATING SEQUENCES IN LOCALLY NORMAL RIESZ SPACES 235

Proof. Let f be a quasi-slowly oscillating continuous function on X and E be a quasi-slowly oscillating
compact subset of X. Let y = (yn) be a sequence of points in f(E). Then we can write yn = f(xn)
where (xn) is sequence of points in E for each n ∈ N. Since E is quasi-slowly oscillating compact,
there is a quasi-slowly oscillating subsequence z = (zk) = (xnk) of (xn). Then, quasi-slowly oscillating
continuity of f implies that f(zk) is a quasi-slowly oscillating subsequence of f(xn). Hence f(E) is
quasi-slowly oscillating compact. �

Corollary 3.6. For any regular subsequential method G, if E is G-sequentially compact subset of X,
then it is quasi-slowly oscillating compact.

Proof. The proof of the result follows from the regularity and subsequence property of G. �

Corollary 3.7. A real valued function defined on a bounded subset of R is uniformly continuous if
and only if it is slowly oscillating continuous.

Proof. The proof of the result follows from the fact that totally boundedness coincides with slowly
oscillating compactness and boundedness coincides with totally boundedness in R. �

Corollary 3.8. Any totally bounded subset of X is quasi-slowly oscillating compact.

Proof. The proof of the result follows from the fact that any sequence of points in a totally bounded
subset of X has a Cauchy subsequence, which is quasi-slowly oscillating. �

Now we give the definition on ideal continuous function in LNRS.

Definition 3.6. Let (X,τ1) and (Y,τ2) be LNR spaces and E ⊂ Y. A function f : E → Y is called
ideally continuous at a point x0 ∈ E if xn

Iτ1→ x0 in E implies f(xn)
Iτ2→ f(x0) in Y.

Theorem 3.9. Let (X,τ1) and (Y,τ2) be LNR spaces. If a function f : X → Y is uniformly continu-
ous, then f is ideally continuous.

Proof. Let f : X → Y be uniformly continuous and xn
Iτ1→ x0 in X. Let θ1 and θ2 be denote the zeros in

X and Y, respectively. Let W be an arbitrary τ2-neighborhood of θ2. Since f is uniformly continuous,
there exists some τ1-neighborhood V of θ1 such that

x−y ∈ V ⇒ f(x) −f(y) ∈ W. (3.1)

Since xn
Iτ1→ x0, we put K = {n ∈ N : xn −x0 /∈ V}, so K ∈ I.

Then from (3.1) we have

f(xn) −f(x0) /∈ W for all n ∈ K.
Therefore we have

{n ∈ N : f(xn) −f(x0) /∈ W}⊆ K
and hence

{n ∈ N : f(xn) −f(x0) /∈ W}∈ I.

i.e. we have f(xn)
Iτ2→ f(x0), which shows that f is ideally continuous. �

Theorem 3.10. A function f on X is ideally sequentially continuous at a point x0 of X if and only
if it is continuous at a point x0 in ordinary sense.

Proof. The proof follows from the fact that any ideally convergent sequence has a convergent subse-
quence and from the above Lemma 3.1. �

Theorem 3.11. Let f : X → X be any function and (xn) be a sequence of points in X such that
Iτ − limn→∞xn = x0 implies limn→∞f(xn) = f(x0), then it is a constant function.

Proof. For the proof of the theorem follows form Theorem 3 in [12]. �

Theorem 3.12. If a function is quasi-slowly oscillating continuous on a subset E of X, then it is
ideally sequentially continuous on E.



236 HAZARIKA AND ESI

Proof. Let f be any quasi-slowly oscillating continuous on E. By Theorem 3.2, we have f is continuous
on E. Also from Theorem 3.10, we see that f is ideally sequentially continuous on E. This completes
the proof. �

Theorem 3.13. If a function is δ-ward continuous on a subset E of X, then it is ideally sequentially
continuous on E.

Proof. Let f be any δ-ward continuous function on E. It follows from Corollary 2 in [11] that f is
continuous. By Theorem 3.10 we obtain that f is ideally sequentially continuous on E. This completes
the proof of the theorem. �

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QUASI-SLOWLY OSCILLATING SEQUENCES IN LOCALLY NORMAL RIESZ SPACES 237

1Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh,
India

2Adıyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adıyaman, Turkey

∗Corresponding author: aesi23@hotmail.com


	1. Introduction
	2. Preliminaries and Notations
	3. Quasi-slowly oscillating sequences in LNRS
	References