International Journal of Science & Technology
Volume 1, No 1, 11-14, 2006
A New Type Of Difference Sequence Spaces
1
Binod CHANDRA TRIPATHY1 and AYHAN ESI2
Mathematical Sciences Division; Institute of Advanced Study in Science and Technology; Paschim Boragaon;
GARCHUK; GUWAHATI-781 035, INDIA
2
Inonu University, Science and Art Faculty in Adiyaman, 02040, Adıyaman, TURKEY
[email protected]
(Received: 17.08.2005; Accepted: 12.12.2005)
Abstract: In this paper we introduce the notion of the difference operator ∆ m x k for a fixed m ∈ N. We define the
sequence spaces l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) (m ∈ N ) and study some topological properties of these spaces.
We obtain some inclusion relations involving these sequence spaces.
2000 AMS . 40 A 05; 40 C 05; 46 A 45.
Key words: Difference sequence space, Solid space, Symmetric space, Completeness
Fark Dizi Uzaylarının Yeni Bir Şekli
Özet: Bu çalışmada sabit bir m ∈ N sayısı için ∆ m x k fark operatörü yardımıyla l ∞ (∆ m ) , c(∆ m ) ve c o (∆ m )
dizi uzayları tanımlanıp, bu uzaylar için bazı topolojik özellikler çalışılmış ve bu uzaylara ait bazı kapsam bağıntıları
verilmiştir.
Anahtar Kelimeler: Fark dizi uzayı, Solid uzay, Simetrik uzay, Tamlık
1. Introduction
2. Definitions and Preliminaries
Throughout the paper w, l ∞ , c, and co
denote the spaces of all , bounded, convergent, and
null sequences x = (xk) with complex terms,
respectively, normed by
|| x ||∞ = supk |xk| .
The zero sequence is denoted by θ = ( 0,0,0, . . ).
Kızmaz [3] defined the difference sequence
spaces l ∞ (∆), c(∆) and co(∆) as follows:
Z (∆) = { x = (xk) : (∆xk ) ∆ Z },
for Z = l ∞ ,c and co , where ∆x = (∆xk) = (xk xk+ı), for all k ∈ N.
The above spaces are Banach spaces, normed by
|| x ||∆ = | x1| + supk || ∆xk || .
The idea of Kizmaz [3] was applied to
introduce different type of difference sequence
spaces and study their different properties by
Tripathy ([6], [7] ), Et and Esi [8] and many
others.
A sequence so ace E said to be solid (or
normal) if (xk) ∈ E implies (αkxk) ∈ E for all
sequences of scalars (αk) with | αk | ≤ 1 for all n∈
N.
A sequence space E is said to be monotone if
it contains the canonical preimages of all its step
spaces.
A sequence space E is said to be convergence
free if (yk) ∈ E whenever (xk) ∈ E and yk = 0
whenever xk = 0.
A sequence space E is said to be a sequence
algebra if (xk.yk) ∈ E whenever (xk) ∈ E and (yk)
∈E.
A sequence space E is said to be symmetric if
(xπ(k)) ∈ E whenever (xk)∈E, where π(k) is a
permutation on N .
For r >0, a nonempty subset V of a linear
space is said to be absolutely r-convex if x, y ∈ V
Tripathy and Esi
and λ
r
r
m
+ µ ≤ 1 together imply that
λx + µy ∈ V. A linear topological space X is said
to be r-convex if every neighborhood of θ ∈X
⇒ xr = 0 for r = 1, 2, - - - , m and
contains as absolutely r-convex neighborhood of
θ ∈X (see for instance Maddox and Roles [5] ).
all k ∈ N.
Consider k = 1 i.e.
x
k
r =1
m
≤
∑| x
m
r
r =1
=
| + sup | ∆ m x k | + ∑ | y r | + sup | ∆ m y k |
k
k
r =1
|| x || ∆ m + || y || ∆ m .
Finally
m
|| λx || ∆ m = ∑ | λx r | + sup | ∆ m (λx k ) |
= |λ|
|| . || ∆ m is a norm on the
spaces l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) .
This completes the proof.
Proposition 2. (i) c o (∆ m ) ⊂ c(∆ m ) ⊂ l ∞ (∆ m ) and
the inclusions are proper.
(ii) Z(∆) ⊂ Z (∆ m ) , for Z = c , c0 , l ∞ and the
inclusions are strict.
m
- - - - - - - (1)
k
Proof. Let α, β be scalars and (xk), (yk) ∈ l ∞ (∆m).
Then
sup | ∆ m x k |< ∞ and sup | ∆ m y k |< ∞
Proof. Trivial.
k
Theorem 3. The sequence spaces
- - - - - - - - - (2)
Hence
l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) are Banach spaces
sup | ∆ m (αx k + βy k ) | ≤ |α| sup | ∆ m x k | + | β|
k
|| x || ∆ m .
Hence
Proposition 1. The classes of sequences
l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) are normed linear
spaces, normed by
= ∑ x r + sup | ∆ m x k |
k
r =1
In this section we establish the results of this
article. The proof of the following result is a routine
verification.
under the norm (1) .
k
(x ) be a Cauchy sequence in l (∆ ) ,
where (x ) = (x ) = (x , x , x ,...) ∈ l (∆ ) for
sup | ∆ m y k | < ∞, by (2).
Proof. Let
k
n
Hence l ∞ (∆m) is a linear space. Similarly it can
be shown that c(∆m) and c0(∆m) are linear spaces.
Next for x = θ, we have || θ || ∆ m = 0 . Conversely,
let
| ∆ m x1 |= 0 ⇒ | x1 –x1+m | =
m
3. Main Results
k
| ∆ m x k |= 0 , for
|| x + y || ∆ m = ∑ | x r + y r | + sup | ∆ m ( x k + y k ) |
For m =1, l ∞ (∆m)= l ∞ (∆), c(∆m)= c(∆) and
c0(∆m)= c0 (∆). Hence the introduced notion generalizes
the notion of difference sequences studied by Kizmaz
[3].
r =1
k
r =1
Proceeding in this way we have xk =0, for all k ∈ N.
Z (∆ m ) = {x = ( x k ) ∈ w : ∆ m x ∈ Z },
for Z = l ∞ ,c and co , where
∆ m x = (∆ m x k ) = ( x k − x k + m ) , for all k ∈ N.
∆m
∆m
⇒ xm+1 = 0, since xm = 0.
Let m ∈ N be fixed, then we introduce the
following new type of difference sequence spaces
x
= ∑ x r + sup | ∆ m x k | = 0.
each n ∈ N .
Then
xn − xi
|| x || ∆ m = 0 .
Then
n
∞
n
i
n
ı
n
2
n
3
∞
m
m
m
∆m
= ∑ x rn − x ri + sup | ∆ m x kn − ∆ m x ki |→ 0
r =1
k
, as n, i → ∞. Hence for a given ε > 0, there exists no ∈
N such that
12
A New Type of Difference Sequence Spaces
Proposition 4. The spaces
m
xn − xi
∆m
= ∑ x rn − x ri + sup | ∆ m x kn − ∆ m x ki |< ε l
, for all n, i ≥ n0.
- - - - (3)
Hence x − x
< ε for all n, i ≥ n0 and
n
k
i
k
(∆ m ) , c(∆ m ) and co (∆ m ) are BK-spaces.
Since the inclusions c(∆ m ) ⊂ l ∞ (∆ m ) and
c o (∆ m ) ⊂ l ∞ (∆ m ) are proper, the following result
k
r =1
follows from Theorem 3.
k = 1,2, . . . , m.
( ) is a Cauchy sequence in C for k = 1,2, . . , m.
⇒ (x ) is a convergent in C for k = 1,2, . . ., m.
⇒ x
i
k
c(∆ m ) and c o (∆ m ) are
nowhere dense subsets of l ∞ (∆ m ) .
Proposition 5. The spaces
i
k
Let
lim x ki = x k , say for k = 1,2, . . . , m.
i →∞
Theorem 6. The spaces
From (3) we have | ∆ m x k − ∆ m x k |< ε , for all
n
i
(
)
l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) are not solid spaces.
n, i ≥ n0 and all k ∈ N. Hence ∆ m x k is a Cauchy
i
(
)
Proof. The proof follows from the following examples.
sequence in C for all k ∈ N. Thus ∆ m x k is
convergent in C, let
i
lim ∆ m x ki = y k , say for each k ∈
Example 1. Let xk = k for all k ∈N. Consider the
sequence of scalars (αk) defined by αk = im +1, for i =
0, 1, 2, . . . and αk = 0, otherwise. Then
(xk)∈ c(∆ m ) ⊂ l ∞ (∆ m ) , but (αkxk) ∉ l ∞ (∆ m ) .
i →∞
N. Since
lim x ki = x k ,exists for k = 1,2, . . . , m , so
we have
lim x ki = x k ,exists for each k ∈ N.
i →∞
c(∆ m ) and l ∞ (∆ m ) are not solid.
For the case c o (∆ m ), consider the sequence xk
i →∞
Hence the spaces
We have
m
m
r =1
r =1
lim ∑ | x ri − x rj |= ∑ | x ri − x r |< ε ,
j →∞
=1 for all k ∈N and the sequence (αk) defined as above.
Then (xk)∈ c o (∆ m ) , but (αkxk) ∉ c o (∆ m ) .
for all i ≥ n0 , and
lim | x ik − x kj − ( x ik + m − x kj + m ) |
j→ ∞
Hence
, for all k ∈N and
Theorem 7. (i) The space
i ≥ n0.
Hence for all i ≥ n0 , we have
(for m > 1) are not symmetric spaces.
sup | ∆x ki − ∆x k |< ε .
k
Proof. (i) The first part is known. For the second part,
consider the following example.
Thus
m
i
r
r =1
c o (∆ m ) is not solid.
c o (∆ ) is symmetric.
(ii) The spaces l ∞ (∆ m ) , c(∆ m ) and c o (∆ m )
=| x ik − x k − ( x ik + m − x k + m ) |< ε
∑| x
∞
− x r | + sup | ∆x ki − ∆x k |< 2ε ,
Example 2. Let m = 2 and consider the sequence (xk)
defined by (xk) = 1 for k odd and (xk) = 2 for k even.
Consider the rearranged sequence (yk) as
k
⇒ ( x − x) ∈ l ∞ ( ∆ m ) , for all i ≥ n0 .
i
Thus x = xi – (xi – x ) ∈
( y k ) = (x1 , x3 , x 2 , x 4 , x5 , x7 , x6 , x8 ,....)
l ∞ (∆ m ) , for all i ≥ n0 ,
l ∞ (∆ m ) is a linear space .
Hence l ∞ (∆ m ) is complete.
since
Then ( y k ) ∉ c 0 (∆ 2 ) . Hence c o (∆ 2 ) is not
symmetric.
Next let m = 1 and consider the sequence (xk) defined as
xk = k for all k ∈ N . Consider its rearrangement defined
Similarly it can be shown that the spaces
c(∆ m ) and c o (∆ m ) are also complete.
The following result is a consequence of the above
result and the definition of BK-space.
13
Tripathy and Esi
as
Example 4. Let m =1 and consider the sequence x =
(xk) defined as xk = 1, for all k ∈ N. Then
(xk)∈ c o (∆ ) . Now consider the sequence (yk) in its
preimage space defined by yk = 1, for k odd and by yk
= 0, for k even, then (yk) ∉ c o (∆ ) . Hence the space
 x1 , x2 , x4 , x3 , x9 , x5 , x16 , x6 , x25 , 

 x7 , x36 , x8 , x49 , x10 ,...

( yk ) = 
c(∆ ) ⊂ l ∞ (∆ ) , but (yk) ∉ l ∞ (∆ ) .
Hence the spaces c(∆ ) and l ∞ (∆ ) are not
Then (xk)∈
c o (∆ ) is not monotone.
l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) are not convergence
Next consider the sequence x = (xk) defined as xk
= k , for all k ∈ N . Then (xk)∈ c ∆ ⊂ l ∞ ∆ . Now
consider the sequence (yk) in its preimage space,
defined as above, then (yk) ∉ l ∞ ∆ . Hence the
free.
spaces
symmetric.
( )
Theorem 8. The spaces
( )
Proof. The result follows from the following example.
c(∆ m ) and l ∞ (∆ m ) are not monotone.
Theorem 10.
convex.
Example 3. Let m = 2 and consider the sequence (xk)
defined as xk = 1, for all k ∈ N . Then (xk)
∈ c o (∆ 2 ) ⊂ c ∆ 2 ⊂ l ∞ ∆ 2 . Now consider the
( )
( )
( )
l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) are 1-
0 p δ p 1 , then
V= x = ( x k ) :
x ∆ ≤δ
Proof. If
{
sequence (yk) defined by yk = k , for all k ∈ N, then
(yk) ∉ l ∞ ∆ 2 . Hence the spaces
2
( )
c o (∆ m ) , c(∆ m ) and l ∞ (∆ m ) are not
m
}
is an absolutely 1-convex set, for let x,y ∈ V and
λ + µ ≤ 1 , then
convergence free .
λ x + µy
Theorem 9. The spaces
l ∞ (∆ m ) , c(∆ m ) and c o (∆ m ) are not monotone.
∆m
≤ ( λ + µ )δ ≤ δ .
This completes the proof.
Proof. The proof follows from the following example.
4. References
6. Tripathy, B.C.(2003). A class of difference
1. Cooke, R.G. (1950). Infinite Matrices and
sequence sequences related to the p-normed space
Sequence Spaces; MacMillan , London.
l p ; Demonstratio Math. ; 36(4), 867-872.
2. Kamthan, P.K. and Gupta, M.(1981):.Sequence
Spaces and Series ; Marcel Dekker Inc. ; New
York.
7. Tripathy, B.C.(2003).On some class of difference
paranormed sequence spaces associated with
multiplier sequences ; Internat. J. Math. Sci.;
vol.2(no.1); 159-166.
3. Kizmaz, H. (1981). On Certain Sequence Spaces,
Canad.Math.Bull.; 24 169-176.
8. Et,M.,.and Esi,A. (2000).On Köthe-Toeplitz duals
4. Maddox, I.J.(1970). Elements of Functional
of generalized difference sequence spaces; Bull.
Malaysian Math.Sci.Soc. (Second Series), 23,
1-8 .
Analysis Cambridge Univ.Press.
5. Maddox I.J.,and Roles, J.W.(1969). Absolute
convexity in certain topological linear spaces,
Proc.Camb.Phil.Soc., 66, 541-545.
14