Gen. Math. Notes, Vol. 1, No. 1, November 2010, pp. 1-7
c
ISSN 2219-7184; Copyright °ICSRS
Publication, 2010
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On n-normed linear space valued
strongly ∇r -Cesàro and strongly
∇r -lacunary summable sequences
B.Surender Reddy1 , Hemen Dutta2 and Iqbal H. Jebril3
1
Department of Mathematics, Osmania University, Hyderabad, A.P., India.
E-mail: bsrmathou@yahoo.com
2
Department of Mathematics, Gauhati University, Kokrajhar Campus, Assam, India
E-mail: hemen− dutta08@rediffmail.com
3
Department of Mathematics, King Faisal University, Saudi Arabia
E-mail: iqbal501@hotmail.com
(Received 2.9.2010, Accepted 1.10.2010)
Abstract
In this article we introduce the spaces |σ1 |(X, ∇r ) and Nθ (X, ∇r ) of Xvalued strongly ∇r -Cesàro summable and strongly ∇r -lacunary summable sequences respectively, where X, a real linear n-normed space and ∇r is a new
difference operator, where r is a non-negative integer. This article extends the
notion of strongly Cesàro summable and strongly lacunary summable sequences
to n-normed linear space valued (n-nls valued) difference sequences. We study
these spaces for existence of norm as well as for completeness. Further we
investigate the relationship between these spaces.
Keywords: n-norm; Difference sequence space; Cesàro summable sequence;
Lacunary summable sequence; Completeness.
2000 MSC No: 46B04, 40A05, 46A45.
2
1
B.Surender Reddy, Hemen Dutta and Iqbal H. Jebril
Introduction
Let w denote the space of all real or complex sequences. By c, c0 and `∞ , we
denote the Banach spaces of convergent, null and bounded sequences x = (xk ),
respectively normed by kxk = sup |xk |.
k
Let Z be a sequence space, then Kizmaz [8] introduced the following sequence spaces: Z(∆) = {x = (xk ) ∈ w : (∆xk ) ∈ Z}, for Z = `∞ , c, c0 , where
∆xk = xk − xk+1 , for all k ∈ N .
The following definitions can be found in [5].
The spaces |σ1 | of strongly Cesàro summable sequence is defined as follows:
|σ1 | = {x = (xk ) : there exists L such that
which is a Banach space
¶ by
µ normed
kxk = sup
p
1
p
p
P
k=1
1
p
p
P
k=1
|xk − L| → 0},
|xk | .
By a lacunary sequence θ = (kp ), p = 1, 2, 3, . . . , where k0 = 0, we mean
an increasing sequence of non-negative integers with hp = (kp − kp−1 ) → ∞ as
kp
p → ∞. We denote Ip = (kp−1 , kp ] and ηp = kp−1
for p = 1, 2, 3, . . .. The space
of strongly lacunary summable sequence Nθ is defined as follows:
P
|xk − L| = 0, for some L}.
Nθ = {x = (xk ) : p→∞
lim h1p
k∈Ip
The space Nθ is a Banach space with the norm
P
kxkθ = sup h1p
|xk |.
p
k∈Ip
The concept of 2-normed spaces was initially developed by Gähler in the
mid of 1960’s, which can be found in [6] while that of n-normed spaces can be
found in [9]. Since then, many others have studied this concept and obtained
various results; see for instance [1, 3, 4, 7].
Definition 1.1 Let n ∈ N and X be a real linear space of dimension d ≥
n ≥ 2. A real valued function k•, •, · · · , •k : X n → R satisfying the following
four properties:
(nN1 ) :
kx1 , x2 , . . . , xn k = 0 if and only if x1 , x2 , . . . , xn are linearly
dependent vectors,
(nN2 ) : kx1 , x2 , . . . , xn k = kxj1 , xj2 , . . . , xjn k for every permutation
(j1 , j2 , . . . , jn ) of (1, 2, . . . , n) i.e., kx1 , x2 , . . . , xn k is invariant under any permutation of x1 , x2 , . . . , xn .
(nN3 ) : kαx1 , x2 , . . . , xn k = |α| kx1 , x2 , . . . , xn k for all α ∈ R
(nN4 ) : kx + x0 , x2 , . . . , xn k ≤ kx, x2 , . . . , xn k + kx0 , x2 , . . . , xn k for all
x, x0 , x2 , . . . , xn ∈ X, is called an n-norm on X and the pair (X, k•, •, · · · , •k)
is called linear n-normed space.
On n-nls valued strongly ∇r -Cesàro and strongly ∇r -lacunary summable
3
The standard n-norm on X, a real inner product space of dimension d ≥ n
is as follows:
¯
¯ < x ,x > ···
1
1
¯
¯
..
..
kx1 , x2 , . . . , xn kS = ¯¯
.
.
¯
¯ < xn , x 1 > · · ·
¯1
< x1 , xn > ¯¯ 2
¯
..
¯
.
¯
¯
< xn , xn > ¯
where < ., . > denotes the inner product on X. If X = Rn , then this n-norm is
exactly the same as the Euclidean n-norm, kx1 , x2 , . . . , xn kE as mention below.
1
For n=1, this n-norm is the usual norm kx1 k =< x1 , x1 > 2 .
A trivial example of an n-normed space is X = Rn equipped with the
following Euclidean n-norm:
¯
¯ x
¯ 11
¯ .

kx1 , x2 , . . . , xn kE = abs ¯¯ ..
¯
¯ xn1
· · · x1n
..
..
.
.
· · · xnn
¯
¯
¯
¯
¯
¯
¯
¯
where xi = (xi1 , . . . , xin ) ∈ Rn for each i = 1, 2, . . . , n.
Definition 1.2 A sequence (xk ) in a linear n-normed space (X, k•, •, · · · , •k)
is said to convergent to L ∈ X if lim kxk − L, w2 , w3 , . . . , wn k = 0, for every
k→∞
w2 , w3 , . . . , wn ∈ X.
Definition 1.3 A sequence (xk ) in a linear n-normed space (X, k•, •, · · · , •k)
is called Cauchy sequence if lim kxk − xm , w2 , w3 . . . , wn k = 0, for every
w2 , w3 , . . . , wn ∈ X.
k,m→∞
Definition 1.4 A linear n-normed space X is said to be complete if every
Cauchy sequence in X is convergent. A complete n-normed space is called an
n-Banach space.
Now we state the following important result [7] on n-norms as Lemma.
Lemma 1.5 A standard n-normed space is complete if and only if it is
1
complete with respect to usual norm k.kS =< ., . > 2 .
2
The Spaces |σ1|(X, ∇r ) and Nθ (X, ∇r )
Throughout this section (X, k•, •, · · · , •kX ) will be a real linear n-normed space
and w(X) will denotes X-valued sequence space. The n-norm k•, •, · · · , •kX
on X is either a standard n-norm or a non-standard n-norm. In general we
write k•, •, · · · , •kX and for standard case we write k•, •, · · · , •kS .
4
B.Surender Reddy, Hemen Dutta and Iqbal H. Jebril
Let r be a non-negative integer. Then we define the following sequence
space:
p
P
|σ1 |(X, ∇r ) = {x ∈ w(X) : p→∞
lim p1
k=1
k∇r xk − L, z1 , z2 , . . . , zn−1 kX = 0, for
every z1 , z2 , . . . , zn−1 ∈ X and for some L}, where ∇r xk = xk − xk−r with
∇0 xk = xk , for all k ∈ N (See for details in [2]). In this expansion, we take
xk = 0, for all non-positive values of k. For L = 0, we write this space as
|σ1 |0 (X, ∇r ).
We call |σ1 |(X, ∇r ), the set of all X-valued strongly ∇r -Cesàro summable
sequences.
Let θ be a lacunary sequence and r be a non-negative integer. Then we
define the following space:
P
Nθ (X, ∇r ) = {x ∈ w(X) : lim h1p
k∇r xk − L, z1 , z2 , . . . , zn−1 kX = 0, for
p→∞
k∈Ip
every z1 , z2 , . . . , zn−1 ∈ X and for some L}. For L = 0, we write this space as
Nθ0 (X, ∇r ).
We call Nθ (X, ∇r ), the set of all X-valued strongly ∇r -lacunary summable
sequences.
In the special case where θ = (2p ), we have Nθ (X, ∇r ) = |σ1 |(X, ∇r ). For
r = 0, we write the above two spaces as |σ1 |(X) and Nθ (X) respectively.
It is obvious that |σ1 |(X) ⊂ |σ1 |(X, ∇r ) and Nθ (X) ⊂ Nθ (X, ∇r ). This
means that every X-valued strongly Cesàro summable sequence is strongly ∇r Cesàro summable and every X-valued strongly lacunary summable sequence
is strongly ∇r -lacunary summable.
Theorem 2.1 (i) If X is an n-Banach space then |σ1 |(X, ∇r ) is a Banach
space normed by
Ã
kxk =
sup
p≥1, z1 ,z2 ,...,zn−1 ∈X
p
1X
k∇r xk , z1 , z2 , . . . , zn−1 kX
p k=1
!
(2.1)
(ii) If X is an n-Banach space then Nθ (X, ∇r ) is a Banach space normed by

kxkθ =
sup
p≥1, z1 ,z2 ,...,zn−1 ∈X


1 X
hp
k∇r xk , z1 , z2 , . . . , zn−1 kX 
(2.2)
k∈Ip
Proof. (i) It is easy to see that |σ1 |(X, ∇r ) is a normed linear space. To prove
completeness, one may use same arguments as applied in [1] and [2].
(ii) Proof of this part follows by applying similar arguments as applied to
prove part (i).
The following Corollary is due to Lemma 1.5.
5
On n-nls valued strongly ∇r -Cesàro and strongly ∇r -lacunary summable
Corollary 2.2 Let X be equipped with standard n-norm. Then
(i) If X is a Banach space then |σ1 |(X, ∇r ) is a Banach space normed by
Ã
kxk =
sup
p≥1, z1 ,z2 ,...,zn−1 ∈X
p
1X
k∇r xk , z1 , z2 , . . . , zn−1 kS
p k=1
!
(ii) If X is a Banach space then Nθ (X, ∇r ) is a Banach space normed by

kxkθ =
p≥1,

1 X

sup
k∇r xk , z1 , z2 , . . . , zn−1 kS 
hp k∈Ip
z1 ,z2 ,...,zn−1 ∈X
Proposition 2.3 Let θ = (kp ) be a lacunary sequence with lim inf ηp > 1,
p
then |σ1 |(X, ∇r ) ⊆ Nθ (X, ∇r ).
Proof. Let lim inf
ηp > 1. Then there exists v > 0 such that 1 + v ≤ ηp for all
p
p ≥ 1. Let x ∈ |σ1 |(X, ∇r ). Then there exists some L ∈ X such that
t
1X
k∇r xk − L, z1 , z2 , . . . , zn−1 kX = 0 for every z1 , z2 , . . . , zn−1 ∈ X
t→∞ t
k=1
lim
1
hp
Now we write
=
1
hp
=
P
1≤i≤kp
kp
hp
Ã
1
kp
P
k∈Ip
k∇r xk − L, z1 , z2 , . . . , zn−1 kX
k∇r xi −L, z1 , z2 , . . . , zn−1 kX − h1p
P
1≤i≤kp
P
1≤i≤kp−1
k∇r xi −L, z1 , z2 , . . . , zn−1 kX
!
k∇r xi − L, z1 , z2 , . . . , zn−1 kX

kp−1  1
−
hp
kp−1

X
k∇r xi − L, z1 , z2 , . . . , zn−1 kX 
(2.3)
1≤i≤kp−1
Now we can have
kp
1+v
kp−1
1
≤
and
≤ , since hp = kp − kp−1
hp
v
hp
v
Hence using (2.3), we have x ∈ Nθ (X, ∇r ).
Proposition 2.4 Let θ = (kp ) be a lacunary sequence with lim sup ηp < ∞,
p
then Nθ (X, ∇r ) ⊆ |σ1 |(X, ∇r ).
6
B.Surender Reddy, Hemen Dutta and Iqbal H. Jebril
Proof. Let lim sup ηp < ∞. Then there exists M > 0 such that ηp < M for
p
all p ≥ 1. Let x ∈ Nθ0 (X, ∇r ) and ² > 0. We can find R > 0 and K > 0 such
that


ki−1
ki
1 X
1 X
sup Si = sup 
k∇r xi , z1 , z2 , . . . , zn−1 kX −
k∇r xi , z1 , z2 , . . . , zn−1 kX  < ε
hi i=1
hi i=1
i≥R
i≥R
and Si < K for all i = 1, 2, . . . .. Then if t is any integer with kp−1 < t ≤ kp ,
where p > R, we can write
kp
t
1X
1 X
k∇r xi , z1 , z2 , . . . , zn−1 kX ≤
k∇r xi , z1 , z2 , . . . , zn−1 kX
t i=1
kp−1 i=1
=
P
1
( k∇r xi , z1 , z2 , . . . , zn−1 kX
kp−1
I1
+
+ ··· +
=
P
I2
P
Ip
k∇r xi , z1 , z2 , . . . , zn−1 kX
k∇r xi , z1 , z2 , . . . , zn−1 kX )
k1
k2 − k1
kR − kR−1
kR+1 − kR
kp − kp−1
S1 +
S2 +. . .+
SR +
SR+1 +. . .+
Sp
kp−1
kp−1
kp−1
kp−1
kp−1
Ã
!
≤ sup Si
i≥1
Ã
kR
kp−1
!
+ sup Si
i≥R
kp −kR
kp−1
R
< K Kkp−1
+ ²M
Since kp−1 → ∞ as t → ∞, it follows that x ∈ |σ1 |0 (X, ∇r ). The general
inclusion Nθ (X, ∇r ) ⊆ |σ1 |(X, ∇r ) follows by linearity.
The following Proposition is the consequence of the above two Propositions.
Proposition 2.5 Let θ = (kp ) be a lacunary sequence with 1 < lim inf
ηp ≤
p
lim sup ηp < ∞, then |σ1 |(X, ∇r ) = Nθ (X, ∇r ).
p
(Note : Part of this paper was presented in ICM-2010, August 19-27, Hyderabad.)
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On n-nls valued strongly ∇r -Cesàro and strongly ∇r -lacunary summable
7
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