Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 81028, 8 pages
doi:10.1155/2007/81028
Research Article
Some Geometric Properties of Sequence Spaces Involving
Lacunary Sequence
Vatan Karakaya
Received 27 August 2007; Accepted 30 October 2007
Recommended by Narendra Kumar K. Govil
We introduce new sequence space involving lacunary sequence connected with Cesaro
sequence space and examine some geometric properties of this space equipped with Luxemburg norm.
Copyright © 2007 Vatan Karakaya. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let (X, · ) be a real Banach space and let B(X)(resp., S(X)) be the closed unit ball (resp.,
the unit sphere) of X. A point x ∈ S(X) is an H-point of B(X) if for any sequence (xn ) in
w
X such that xn →1 as n→∞, weak convergence of (xn ) to x (write xn → x) implies that
xn − x→0 as n→∞. If every point in S(X) is an H-point of B(X), then X is said to have
the property (H). A point x ∈ S(X) is an extreme point of B(X) if for any y,z ∈ S(X) the
equality x = (y + z)/2 implies y = z. A point x ∈ S(X) is a locally uniformly rotund point
of B(X)(LUR-point) if for any sequence (xn ) in B(X) such that xn + x→2 as n→∞,
there holds xn − x→0 as n→∞. A Banach space X is said to be rotund (R) if every point
of S(X) is an extreme point of B(X). If every point of S(X) is an LUR-point of B(X),
then X is said to be locally uniformly rotund (LUR). If X is LUR, then it has R-property.
For these geometric notions and their role in mathematics, we refer to the monograph
[1–10].
By a lacunary sequence θ = (kr ) where k0 = 0, we will mean an increasing sequence of
nonnegative integers with kr − kr −1 →∞ as r →∞. The intervals determined by θ will be
denoted by Ir = (kr −1 ,kr ]. We write hr = kr − kr −1 . The ratio kr /kr −1 will be denoted by qr .
The space of lacunary strongly convergent sequences Nθ was defined by Freedman et al.
[12] as
2
Journal of Inequalities and Applications
Nθ = x = x k
1 xk − = 0, for some .
: lim
r →∞ hr
k ∈I
(1.1)
r
It is well known that there exists very close connection between the space of lacunary strongly convergent sequences and the space of strongly Cesaro summability sequences. One can find this connection in [11–16]. Because of these connections, a lot
of geometric property of Cesaro sequence spaces can generalize the lacunary sequence
spaces.
Let w be the space of all real sequences. Let p = (pr ) be a bounded sequence of the
positive real numbers. We introduce the new sequence space l(p,θ) involving lacunary
sequence as follows:
l(p,θ) =
∞
x = xk :
r =1
1 x k h r k ∈I
pr
<∞ .
(1.2)
r
Paranorm on l(p,θ) is given by
xl(p,θ) =
∞
r =1
1 x k h r k ∈I
pr 1/H
,
(1.3)
r
where H = supr pr . If pr = p for all r, we will use the notation l p (θ) in place of l(p,θ). The
norm on l p (θ) is given by
xl p (θ) =
∞
r =1
1 x k h r k ∈I
p 1/ p
.
(1.4)
r
It is easy to see that the space l(p,θ) with (1.3) is a complete paranormed space.
By using the properties of lacunary sequence in the space l(p,θ), we get the following
sequences. If θ = (2r ), then l(p,θ) = ces(p). If θ = (2r ) and pr = p for all r, then l(p,θ) =
ces p .
For x ∈ l(p,θ), let
σ(x) =
∞
r =1
1 x k h r k ∈I
pr
(1.5)
r
and define the Luxemburg norm on l(p,θ) by
x
x = inf ρ > 0 : σ
≤1 .
ρ
(1.6)
Vatan Karakaya 3
The Luxemburg norm on l p (θ) can be reduced to a usual norm on l p (θ), that is, xl p (θ) =
x. To do this, we have
x
≤1
x = inf ρ > 0 : σ
ρ
= inf ρ > 0 :
= inf ρ > 0 :
∞
r =1
∞
=
∞
r =1
r =1
= inf ρ > 0 :
∞
1 xk h r k ∈I ρ
r
1 xk h r k ∈I
r =1
p
≤1
p
≤ρ
r
1 x k h r k ∈I
p
(1.7)
p 1/ p
≤ρ
r
1 x k h r k ∈I
p 1/ p
= xl p (θ) .
r
The main purpose of this work is to show that the space l(p,θ) equipped with Luxemburg norm is a modular space and to investigate the geometric structure of this space.
2. Main results
In this section, first we give some theorems which show the connection between l(p,θ)
and ces(p).
Theorem 2.1. If lim inf qr > 1, then ces(p) ⊂ l(p,θ).
Proof. If lim inf qr > 1, then there exists δ > 0 such that qr > 1 + δ for all r ≥ 2. Then for
x ∈ ces(p), we have
∞
r =2
1 x i hr i∈Ir
pr
=
∞
r =2
≤C
k
k
r −1
1 r x i − 1
x i h r i =1
hr i=1
∞
k
r =2
1 r x i h r i =1
r =2
hr kr i=1
pr
+
∞
pr
r =2
k
r −1
1
x i h r i =1
pr (2.1)
pr ∞ pr k
∞
r −1
kr −1 1 k
kr 1 r xi
+
xi
,
=C
hr kr −1 i=1
r =2
where C = max (1,2H −1 ). Since hr = kr − kr −1 , we have
kr
δ
<
,
hr 1 + δ
kr −1 1
< .
hr
δ
(2.2)
By using (2.2), we have
∞
r =2
1 x i hr i∈Ir
pr
≤C
∞
r =2
k
1 r x i kr i=1
pr
+
∞
r =2
kr −1
1 x i kr −1 i=1
pr .
(2.3)
4
Journal of Inequalities and Applications
Since x ∈ ces(p), we get that
∞
r =2
∞
k
1 r x i kr i=1
pr
< ∞,
pr
kr −1
1 x i kr −1 i=1
r =2
(2.4)
< ∞.
So we obtain that x ∈ l(p,θ).
Theorem 2.2. If 1 < lim sup qr < ∞ , then l(p,θ) ⊂ ces(p).
Proof. We suppose that 1 < lim sup qr < ∞, then there exists positive number K such that
1 < qr < K for all r ≥ 2. Then if m is any integer with kr −1 < m ≤ kr and x ∈ l(p,θ), we
can write
∞
m=1
m
1 x i m i=1
m
1 x i m i =1
pr
≤
k
1 r x i pr
kr −1 i=1
pm
≤C
∞
r =2
≤C
∞
r =2
,
kr −1
1 x i kr −1 i=1
kr −1
1 x i pr
+
r =2
pr
kr −1 i=1
∞
+
∞
r =2
1 x i pr (2.5)
kr −1 i∈Ir
hr 1 x i kr −1 hr i∈Ir
pr .
Since hr /kr −1 = (kr − kr −1 )/kr −1 = qr − 1 < K − 1, we get l(p,θ) ⊂ ces(p).
Now we give some lemmas about convex modular on l(p,θ)
Lemma 2.3. The functional σ is a convex modular on l(p,θ)
Proof. It is clear that σ(x) = 0 ⇔ x = 0 and σ(αx) = σ(x) for all scalars α with |α| = 1. Let
α ≥ 0 and β ≥ 0 with α + β = 1. By the convexity of |t |→|t | pr for every r ∈ N, we have
σ αx + βy =
∞
r =1
≤α
∞
r =1
pr
1 αx(i) + βy(i)
hr i∈Ir
pr
1 x(i)
hr i∈Ir
+β
∞
r =1
pr
1 y(i)
hr i∈Ir
(2.6)
= ασ(x) + βσ(y).
Lemma 2.4. For x ∈ l(p,θ) , the modular σ on l(p,θ) satisfies the following properties:
(i) if 0 < a < 1, then aH σ(x/a) ≤ σ(x) and σ(ax) ≤ aσ(x);
(ii) if a > 1, then σ(x) ≤ aH σ(x/a);
(iii) if a ≥ 1, then σ(x) ≤ aσ(x/a) ≤ σ(ax).
Vatan Karakaya 5
Proof. (i) Let 0 < a < 1. Then we have
σ(x) =
=
∞
r =1
∞
a
1 x(i)
hr i∈Ir
pr
pr
=
∞
r =1
p r
1 x(i) a hr i∈Ir a p r
x
1 H
x(i) ≥
a
σ
.
a h
a
(2.7)
r i∈Ir
r =1
The property σ(ax) ≤ aσ(x) follows from the convexity of σ
(ii) Let a > 1. Then we have
σ(x) =
=
∞
r =1
∞
a
1 x(i)
hr i∈Ir
pr
pr
=
∞
r =1
p r
1 x(i) a hr i∈Ir a p r
x
1 x(i) ≤ aH σ
.
h
a
a
(2.8)
r i∈Ir
r =1
(iii) follows from the convexity of σ.
By the following lemma, we give some connections between the modular σ and the
Luxemburg norm on l(p,θ).
Lemma 2.5. For any x ∈ l(p,θ),
(i) if x < 1, then σ(x) ≤ x;
(ii) if x > 1, then σ(x) ≥ x;
(iii) x = 1 if and only if σ(x) = 1;
(iv) x < 1 if and only if σ(x) < 1;
(v) x > 1 if and only if σ(x) > 1.
Proof. (i) Let ε > 0 be such that 0 < ε < 1 − x. Then we obtain that ε + x < 1. By
definition of norm, there exists ρ > 0 such that ε + x > ρ and σ(x/ρ) ≤ 1. By (i) and (iii)
of Lemma 2.4, we have
σ(x) ≤ σ
x
ε + x x
= σ ε + x ρ
ρ
x
≤ ε + x .
≤ ε + x σ
ρ
(2.9)
Hence, we obtain that σ(x) ≤ x, and (i) is satisfied.
(ii) If x > 1, then 0 > 1 − x and 0 > (1 − x)/ x. Hence, we get that (x −
1)/ x > 0. Let ε > 0 be such that 0 < ε < (x − 1)/ x. Since (x − 1)/ x > 0 and
x(ε − 1) < −1, we can write −1/(x(ε − 1)) < 1 < 1/(x(ε − 1)). By definition of .
and Lemma 2.4(i), we have
x
1
σ(x).
≤
1<σ
(1 − ε)x
(1 − ε)x
So (1 − ε)x ≤ σ(x) for all ε ∈ (0,(x − 1)/ x), which implies that x ≤ σ(x).
(2.10)
6
Journal of Inequalities and Applications
(iii) Assume that x = 1. Let ε > 0, then there exists ρ > 0 such that 1 + ε > ρ > x
and σ(x/ρ) ≤ 1. By Lemma 2.4(i), we have σ(x) ≤ ρH σ(x/ρ) ≤ σ(x/ρ) ≤ ρH < (1 + ε)H ,
so (σ(x))1/H ≤ 1 + ε for all ε > 0 which implies that σ(x) ≤ 1. If σ(x) < 1, let a ∈ (0,1)
such that σ(x) < aH < 1. From Lemma 2.4(i), we have σ(x/a) ≤ (1/aH )σ(x) ≤ 1, hence
x ≤ a < 1, which is a contradiction. Thus, we have σ(x) = 1.
Conversely, assume that σ(x) = 1, by the definition of . we get that x ≤ 1. If x ≤
1, then by (i), we have that σ(x) < x, which contradicts to our assumption, so we obtain
that x = 1.
(iv) follows from (i) and (iii).
(v) follows from (iii) and (iv).
Lemma 2.6. For x ∈ l(p,θ),
(i) if 0 < a < 1 and x > a, then σ(x) > aH ;
(ii) if a ≥ 1 and x < a, then σ(x) < aH .
Proof. (i) We suppose that 0 < a < 1 and x > a. Then x/a > 1. By Lemma 2.5(ii), we
have σ(x/a) > x/a > 1. Hence, by Lemma 2.4(i), we get that σ(x/a) ≥ aH σ(x/a) > aH .
(ii) We suppose that a > 1 and x < a. Then x/a < 1. By Lemma 2.5(i), σ(x/a) <
x/a < 1. If a = 1, we have σ(x) < 1, by Lemma 2.4(ii), we obtain that σ(x) < aH σ(x/a) <
aH .
Lemma 2.7. Let (xn ) be a sequence in l(p,θ),
(i) if lim n→∞ xn = 1, then lim n→∞ σ(xn ) = 1;
(ii) if lim n→∞ σ(xn ) = 0, then lim n→∞ xn = 0.
Proof. (i) Suppose that lim n→∞ xn = 1. Let ε ∈ (0,1). Then there exists n0 such that
1 − ε < xn < 1 + ε for all n ≥ n0 . By Lemma 2.6, (1 − ε)H < xn < (1 + ε)H for all n ≥ n0 ,
which implies that lim n→∞ σ(xn ) = 1.
(ii) Suppose that xn 0. Then there is an ε ∈ (0,1) and subsequence (xnk ) of (xn )
such that xnk > ε for all k ∈ N. By Lemma 2.6(i), we obtain that σ(xnk ) > εH for all
k ∈ N. This implies σ(xnk ) 0 as n→∞.
Lemma 2.8. Let (xn ) be a sequence in l(p,θ). If σ(xn )→σ(x) as n→∞ and xn (i)→x(i) as
n→∞ for all i ∈ N , then xn →x as n→∞.
Proof. Let ε > 0 be given. We put that
σ 0 (x) =
σ 1 (x) =
r0
r =1
pr
1 x(i)
hr i∈Ir
∞
r =r0 +1
,
(2.11)
pr
1 x(i)
hr i∈Ir
.
Since σ(x) < ∞, there exists r0 ∈ N such that
σ 1 (x) <
ε 1
.
3 2H+1
(2.12)
Vatan Karakaya 7
Again, since σ(xn ) − σ 0 (xn )→σ(x) − σ 0 (x) as n→∞, there exists n0 ∈ N such that
σ 1 xn = σ xn − σ 0 xn ≤ σ(x) − σ 0 (x) +
ε 1
.
3 2H+1
(2.13)
Also since xn (i)→x(i) as n→∞ for all i ∈ N, we can take
σ 0 xn − x ≤
ε
3
(2.14)
for all n ≥ n0 . It follows from (2.12), (2.13), and (2.14) for all n ≥ n0 ,
ε
+ 2H σ 1 xn + σ 1 (x)
3
ε 1
ε
H
+
σ
(x)
≤ + 2 σ(x) − σ 0 (x) +
1
3
3 2H
ε 1
ε
= + 2H 2σ 1 (x) +
3
3 2H
ε 1
ε
ε 2
+
≤ + 2H
= ε.
3
3 2H+1 3 2H
σ xn − x = σ 0 xn − x + σ 1 xn − x ≤
(2.15)
This show that σ(xn − x)→0 as n→∞. Hence, by Lemma 2.7(ii), we get that xn − x→0
as n→∞.
Theorem 2.9. The space l(p,θ) has the property (H).
w
Proof. Let x ∈ S(l(p,θ)) and xn (i) ⊆ l(p,θ) such that xn (i)→1 and xn (i) → x(i) as n→∞.
From Lemma 2.5(iii), we get σ(x) = 1. So from Lemma 2.6(i), it follows that σ(xn )→σ(x)
as n→∞. Since mapping π i : l(p,θ)→R defined by π i (yi ) = y(i) is a continuous linear
functional on l(p,θ). It follows that xn (i)→x(i) as n→∞ for all i ∈ N.
Corollary 2.10. For 1 ≤ p < ∞, the space l p (θ) with respect to Luxemburg norm has Hproperty.
Proof. The proof is obtained directly form Theorem 2.9.
Remark 2.11. For a bounded sequence of positive real numbers p = (pr ) with pr ≥ 1
for all r ∈ N, the space l(p,θ) equipped with the Luxemburg norm is not rotund, so
it is not LUR. In [9], it is shown that the space ces(p) equipped with the Luxemburg
norm is not rotund nor LUR. Since ces(p) ⊂ l(p,θ) from Theorem 2.1, we obtain that
the space l(p,θ) has neither R-property nor LUR property. Furthermore, if we take lacunary sequence θ = (kr ) = {2r −1 , r even; 2r ,r odd} and x = {0,0,0,0,0,2,3,0,0,0,... },
y = {1,1,0,0,0,0,0,0,... }, we get that the space l(p,θ) is not rotund.
Indeed, we take x, y ∈ S(l(p,θ)) such that σ(x) = σ(y) = 1. Since σ((x + y)/2)=1, we
have (x + y)/2=1 by Lemma 2.5(iii). This shows that l(p,θ) is not rotund, so it is not
LUR.
8
Journal of Inequalities and Applications
References
[1] B.-L. Lin, P.-K Lin, and S. L. Troyanski, “Some geometric and topological properties of the unit
sphere in Banach spaces,” Mathematische Annalen, vol. 274, no. 4, pp. 613–616, 1986.
[2] S. T. Chen, “Geometry of Orlicz spaces,” Dissertationes Mathematicae, vol. 356, pp. 204, 1996.
[3] Y. A. Cui and H. Hudzik, “On the Banach-Saks and weak Banach-Saks properties of some Banach sequence spaces,” Acta Scientiarum Mathematicarum, vol. 65, no. 1-2, pp. 179–187, 1999.
[4] Y. Cui, H. Hudzik, M. Nowak, and R. Płuciennik, “Some geometric properties in Orlicz sequence
spaces equipped with Orlicz norm,” Journal of Convex Analysis, vol. 6, no. 1, pp. 91–113, 1999.
[5] J. Diestel, Geometry of Banach Spaces-Selected Topics, Springer, Berlin, Germany, 1984.
[6] M. A. Japón, “Some geometric properties in modular spaces and application to fixed point theory,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 576–594, 2004.
[7] J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics,
Springer, Berlin, Germany, 1983.
[8] N. Petrot and S. Suantai, “On uniform Kadec-Klee properties and rotundity in generalized Cesàro sequence spaces,” International Journal of Mathematics and Mathematical Sciences,
vol. 2004, no. 1–4, pp. 91–97, 2004.
[9] W. Sanhan and S. Suantai, “Some geometric properties of Cesaro sequence space,” Kyungpook
Mathematical Journal, vol. 43, no. 2, pp. 191–197, 2003.
[10] W. Sanhan, A. Kananthai, M. Musarleen, and S. Suantai, “On property (H) and rotundity of
difference sequence spaces,” Journal of Nonlinear and Convex Analysis, vol. 3, no. 3, pp. 401–409,
2002.
[11] G. Das and B. K. Patel, “Lacunary distribution of sequences,” Indian Journal of Pure and Applied
Mathematics, vol. 20, no. 1, pp. 64–74, 1989.
[12] A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,” Proceedings of the London Mathematical Society, vol. 37, no. 3, pp. 508–520, 1978.
[13] J. A. Fridy and C. Orhan, “Lacunary statistical summability,” Journal of Mathematical Analysis
and Applications, vol. 173, no. 2, pp. 497–504, 1993.
[14] V. Karakaya, “On lacunaryσ-statistical convergence,” Information Sciences, vol. 166, no. 1–4, pp.
271–280, 2004.
[15] M. Mursaleen and T. A. Chishti, “Some spaces of lacunary sequences defined by the modulus,”
Journal of Analysis, vol. 4, pp. 153–159, 1996.
[16] S. Pehlivan and B. Fisher, “Lacunary strong convergence with respect to a sequence of modulus functions,” Commentationes Mathematicae Universitatis Carolinae, vol. 36, no. 1, pp. 69–76,
1995.
Vatan Karakaya: Department of Mathematics, Education Faculty, Adıyaman University,
3002 Adıyaman, Turkey
Email addresses: vkkaya@yahoo.com; vkarakaya@adiyaman.edu.tr