Some Vector Fk Sequence Spaces Generated by Modulus Function

N. Irsyad

, E. Rosmani

and Herawati

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Medan, Indonesia

Keywords: Vector Value Sequence Space, Modulus Function, Paranorm.

Abstract: In this paper, some vector value sequence spaces G

(X) and L

(X) using modulus function are presented.

Furthermore, we examined some topological properties of these sequence spaces equipped with a paranorm.

1 INTRODUCTION

Let X be a vector space and R be the set of real

numbers. A function f : R

[ f0g ! R

[ f0g is called

modulus function if following condition of f

satisfying:

1. f is vanishing at zero

2. f satisfies triangle inequality

3. f is an increasing function i.e. f ( ) "

The function f must be continuous for every ele-

ment x in (0;¥). The space of all real number se-

quences (x

) such that the infinite series of absolute

modulus function is finited denoted by `( f ) (Ruckle,

1973)

1. f (jx

j) < ¥

n=1

The space `( f ) becomes a FK-space under the F-

norm.

p(x) = å f (jx

j) < ¥

n=1

(Adnan, 2006) examined the FK-space proper-

ties of an analytic and entire real sequence space

using modulus function. (Adnan, 2006) showed the

characterization to matrix transformation of Ruckle’s

space `( f ) into analytic FK-space. For the theory of

FK-space we refer to Banas and Mursaleen[2].

Through the article W(X), G

(X), L

(X) denoted

by the space of vector value sequences, entire vector

value sequence space and analytic vector value

sequence space. The vector value sequence space

studied by some authors (Herawati et al., 2016;

Gultom et al., 2018; Kolk., 2011; Leonard., 1976;

Das and Choudhury., 1992; Et., 2006; Et et al., 2006;

Tripathy et al., 2004; Tripathy et al., 2003), and many

others. Further, the concept of sequence space using

modulus function was investigated by (Bilgin., 1994;

Pehlivan and Fisher., 1994; Waszak., 2002;

Bhardwaj., 2003, Altin., 2009), and many others.

Recently, (Herawati et al., 2016) studied the

geometric of the vector value sequence spaces

defined by order-j function under Lattice norm.

Further, (Gultom et al 2018) studied some topologies

properties of a finite arithmatic mean vector value

sequence space denoted by W

(X) for X is a linear

space and f is a j-function.

A functional is called paranormed if satisfies the

properties p : X ! R that satisfies the properties p(q)

= 0, with q is the zero vector in X, non-negative, p

satisfies triangle inequalities, even and every real

sequence (l

) with jl

lj ! 0. The space X with

paranorm p is called paranormed space, written as

X = (X; p). (Nakano, 1951; Simons, 1965)

In this work, we define the space of vector value

sequences G

(X) and L

(X) called entire and analytic

vector valued sequence spaces generated by modulus

function and study the topological properties of the

sets equipped with paranorm.

Irsyad, N., Rosmani, E. and Herawati, .

Some Vector FK Sequence Spaces Generated by Modulus Function.

DOI: 10.5220/0010079509750978

In Proceedings of the International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches (ICOSTEERR 2018) - Research in Industry 4.0, pages

975-978

ISBN: 978-989-758-449-7

975

2 MAIN RESULTS

In this main result section, firstly, we introduce para-norm

on this space and examine some topological properties

such as complete properties. Let X be a Banach space and

f be a modulus function. Let y(n) = f (kx(n)k

) 2 R for all

natural numbers n, then we get a sequence y = (y(n)). We

define the sets

(X) = x = (x(n))

n2N

: x(n) 2 X and (y(n)) n ! 0;

n ! ¥

(X) = x = (x(n))n 2 N : x(n) 2 X and

sup f(y(n)) n g < ¥

n2N

Theorem 2.1. The sets G

(X) and L

(X) are vector

spaces.

Proof. Let x;z be any elements in G

(X), then

1 1

lim (y(n)) n = 0 and lim (w(n)) n = 0

n!¥ n!¥

for n ! ¥, with y(n) = f (x(n) and w(n) = f (z(n))

for each natural number n. We will apply the

following inequality : if a

2 R and 0 q

sup q

H for each natural number n, then

+ b

M(ja

) + jb

where M = maxf1;2

g. Therefore,

(

(n) + w(n)) n

(y(n)) n + (w(n))

Since (q

)

= (

), then H =

sup

= 1.

Thus

1 1

(

(n) + w(n)) n

(y(n)) n + (w(n))

for n ! ¥, then

Since (y(n)) n ! 0 and (w(n))

(y(n) + w(n)) n ! 0 for n ! ¥. Therefore, we ob-

tain x + y 2 G

(X). Further, for element x 2 G

(X)

and a 2 R, then

(y(n)) n ! 0;n ! ¥

Because of an increasing function f and the

positivity of jaj, then from the Archimedean

properties, there exists natural number n

with

f (jajkx(n)k) f (2

kx(n)k)

Since f satisfies 4

-condition, we get

1 0 1

( f (2

kx(n)))

(

(kx(n)k))

for each natural number n. It shows that ax 2 G

(X). Because x + z 2 G

(X) and ax 2 G

(X) for each

x;y 2G

(X) and each a 2 R, we get G

(X) is a vector

or lin-ear space and the proof of the theorem is

finished. In the same way, it can be shown that L

(X) is a vector space.

Theorem 2.2. A functional p : G

(X) ! R defined by

p(x) = sup

(y(n))

is a paranorm.

Proof. Let x be an element in G

(X). It is clear that

the functional p is non-negative, p(q) = 0, with q is the

zero vector in X and even, for each x 2 G

(X). Now, we

will show that p satisfies the triangle in-equality. To do

that, take any x;z 2 G

(X), then

lim (y(n))

n = 0 and lim (w(n)) n = 0

n!¥ n!¥

for n ! ¥, with y(n) = f (x(n)) and w(n) = f (z(n))

for each n 2 N. we obtain

(y(

)

(

))

(y(

))

(

))

1 1 1

Therefore, there’s vector sequences of x;y 2 G

(X), we get p satisfies the triangle inequality. Next,

we will show that p satisfies the continuity of scalar

mul-tiplication. To do that, take any real sequence

) and (x(n)) 2 G

(X) with jl

lj ! 0 for n 2 ¥. We

have

1 1

( f (kx(n))k

)) n =( f (kl

x(n) lx(n)k)) n

=( f (k(l

l)x(n)k) +

kl(x(n) x)k)) n

(( f jl

ljkx(n)k +

jljk(x(n) x)k) n

ICOSTEERR 2018 - International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches

976

p(l

x(n) lx(n)) = sup f( f (kl

x(n) lx(n)k)) n g

ljp(x(n) + jljp(x(n) x) ! 0

Hence, p(l

x(n) lx(n)) ! 0. The proof of the the-

orem is finished.

Theorem 2.3. The vector spaces of G

(X) and L

(X) are complete paranormed sequence space under the

paranorm defined in Theorem 2.2.

Proof. Take any Cauchy sequence (x

) in G

(X)

with x

= (x

(n)) = (x

(1);x

(2);:::;). Therefore, for

any positive real number e, there exists i

2 N, for all

j i i

, we get

)

(

)

(

)

))

n < e

< e, we have

Since sup ( f kx

(n) x

(n)k)) n

( f (kx

(n) x

(n)k))

< e for e > 0. Since f is a mod-

ulus function, then kx

(n) x

(n)k = 0 for each nat-

ural number n. In other words, kx

(n) x

(n)k < e.

It shows that for each natural number n of the se-

quence (x

(n)) is a Cauchy. Since X is a complete

normed space, then (x

(n)) converges to x(n) 2 X.

Hence, lim x

(n) = x(n) for all n. Therefore, there’s

j!¥

sequence x = (x(n)) = (x(1);x(2);:::;) such that

i!¥

(

( x

(

f (

m x

)

sup

lim(

( x

)

= lim sup

(

( x

)

for every i i

. By using the definition of para-

norm, we get

p(x x

) =

sup

(

(kx

))

It shows that x

! x for i ! ¥. Then it will be shown

that x 2 G

(X). Using the continuous property of f ,

we get

(

( x

)

= (

f (

m x

)

= lim( f ( x

)

for i ! ¥. Hence, x 2 G

(X). The proof of this

theorem is finished.

3 CONCLUSIONS

According to the main results, it can be concluded

(X) and L

(X) are complete paranormed sequence

space under the paranorm.

ACKNOWLEDGEMENTS

Authors would like to say thank you to Talenta USU

2018 for the financial support to join the conference

and also to the reviewers for the revision of this

paper.

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