1
Acta Math. Univ. Comenianae
Vol. LXXIX, 1(2010), pp. 1–8
SOME GEOMETRIC PROPERTIES OF A GENERALIZED
CESÀRO SEQUENCE SPACE
V. A. KHAN
Abstract. In this paper we define the generalized Cesàro sequence space Cesp (q)
and exhibit some geometrical properties of the space. The results herein proved are
exhibit analogous to those by Y. A. Cui [Southeast Asian Bull. Math. 24 (2000),
201–210] for the Cesàro sequence space Cesp .
1. Introduction
Let (X, k.k) be a real Banach space. By B(X) and S(X), we denote the closed unit
ball and the unit sphere of X, respectively. For any subset A of X, χA represents
a characteristics function of A.
A norm k.k is called uniformly convex (UC) (cf. [2]) if, for each ε > 0, there
exists δ > 0 such that, for x, y ∈ S(X), kx − yk > ε implies
1
(x + y) < 1 − δ.
(1.0.1)
2
A Banach space X is said to have the Banach-Saks property if every bounded
sequence (xn ) in X admits a subsequence {zn } such that the sequence {tk (z)} is
convergent in norm in X (see [1]), where
1
(z1 + z2 + · · · + zk ).
k
Every Banach space X with the Banach-Saks property is reflexive, but the
converse is not true (see [4, 5]). Kakutani [6] proved that any uniformly convex
Banach space X has the Banach-Saks property. Moreover, he also proved that if
X is a reflexive Banach space and θ ∈ (0, 2) such that for every sequence (xn ) in
S(X) weakly convergent to zero, there exist n1 , n2 ∈ N satisfying the Banach-Saks
property.
For a sequence (xn ) ⊂ X, we define
tk (z) =
A(xn ) = lim inf{kxi + xj k : i, j ≥ n, i 6= j}.
n→∞
Received August 10, 2005; revised November 02, 2009.
2000 Mathematics Subject Classification. Primary 46E30, 46E40, 46B20.
Key words and phrases. Banach-Saks property; ε-separated sequence; property β; Generalized Cesàro sequence space.
2
V. A. KHAN
In [6], the following new geometric constant connected with the packing constant (see [7]) and with the Banach-Saks property was defined
C(X) = sup{A(xn ) : (xn ) is a weakly null sequence in S(X)}.
Recall that a sequence (xn ) is said to be an ε-separated sequence if, for some ε > 0,
sep(xn ) = inf{kxn − xm k : n 6= m} > ε.
A Banach space X is said to satisfy property (β) if and only if, for every ε > 0,
there exists δ > 0 such that, for each element x ∈ B(X) and each sequence (xn )
in B(X) with sep(xn ) ≥ ε, there is an index k such that
x + xk for some k ∈ N.
2 | ≥ 1 − δ,
In this paper, we define the generalized Cesàro sequence space as follows: let
p ∈ [1, ∞) and q be a bounded sequence of positive real numbers such that
Qn =
n
X
qk ,
(n ∈ N),
k=0
Cesp (q) =


x = (x(i)) : kxk =

∞
X
n=1

!p !1/p
n

X
1
|qi x(i)|
<∞ .

Qn i=1
If qi = 1 for all i ∈ N, then Cesp (q) is reduced to Cesp (cf. [3, 8, 9]).
Lemma 1.1. Let x, y ∈ Cesp (q). Then for any ε > 0 and L > 0, ∃ δ > 0 such
that
| kx + ykp − kxkp | < ε
whenever
kxkp ≤ L
and
kykp ≤ δ.
Proof. For any fix ε > 0 and L > 0, take β = 2−p L−1 ε and δ = 2−p β p−1 ε.
Then for any x, y ∈ Cesp (q) with kxkp ≤ L and kykp ≤ δ, we have
∞
n
X
1 X
(
|qi x(i) + qi y(i)|)p
Q
n
n=1
i=1
!p
∞
n X
qi y(i) 1 X =
(1 − β)qi x(i) + β(qi x(i) +
)
Qn i=1 β n=1
kx + ykp =
!p
n
n
1 X
1 X
qi y(i)
≤
(1 − β)
|qi x(i)| + β
|qi x(i) +
|
Qn i=1
Qn i=1
β
n=1
!p
!p
∞
n
∞
n
X
X
1 X
1 X
qi y(i)
≤ (1 − β)
|qi x(i)| + β
|qi x(i) +
|
Qn i=1
Qn i=1
β
n=1
n=1
∞
X
3
GENERALIZED CESÀRO SEQUENCE SPACE
!p
n
∞
βX
1 X
≤
|qi x(i)| +
Qn i=1
2 n=1
n=1
!p
∞
n
βX
1 X 2qi y(i)
+
|
|
2 n=1 Qn i=1
β
∞
X
p
≤ kxk + ε/2 + (2/β)
p−1
∞
X
n=1
n
1 X
|2qi x(i)|
Qn i=1
!p
!p
n
1 X
|qi y(i)|
Qn i=1
≤ kxkp + ε.
2. Main Results.
Theorem 2.1. The space Cesp (q) satisfies the property (β).
Proof. Let Cesp (q) not have property (β). Then there exists ε0 > 0 such that,
for any δ ∈ (0, ε0 /(1+21+p )), there is a sequence (xn ) ⊂ S(Cesp (q)) with sep(xn ) >
1/p
ε0 and an element x0 ∈ S(Cesp (q)) such that
xn + x0 p
>1−δ
for any n ∈ N.
2
Fix δ ∈ (0, ε0 /(1 + 21+p )). We want to show that
!p
∞
n
X
1 X
21+p δ
lim sup
(2.1.1)
|qi xk (i)|
≤ p
.
j→∞ k
Qn i=1
2 −1
n=j+1
Otherwise, without loss of generality, we can assume that there exists a sequence
(jk ) such that jk → ∞ as k → ∞ and
!p
∞
n
X
1 X
21+p δ
(2.1.2)
|qi xk (i)|
> p
for every k ∈ N.
Qn i=1
2 −1
n=j +1
k
Let δ > 0 be a real number corresponding to ε = δ and L = 1 in Lemma 1.
By absolute continuity of the norm of x0 , there exists a positive integer n1 such that
!p
∞
n
X
1 X
p
|qi x0 (i)|
kx0 χn1 ,n1 +1,n1 +2,... k =
< δ1 .
Qn i=1
n=n +1
1
Choose k so large that jk > n1 . By the Lemma 1 and (2.1.2), we have
!p
n ∞
X
1 X qi xk (i) + qi x0 (i) 1−δ <
Qn i=1 2
n=1
!p
!p
n1
n ∞
n X
X
1 X qi xk (i)+qi x0 (i) 1 X qi xk (i) + qi x0 (i) =
+
Qn i=1 2
Qn i=1 2
n=n +1
n=1
1
4
V. A. KHAN
!p
!p
n1
n
n
1 X
1 X
1X
|qi x0 (i)| +
|qi xk (i)|
Qn i=1
2 n=1 Qn i=1
!p
n ∞
X
1 X qi xk (i) +δ
+
Qn i=1 2 n=n1 +1
!p
!p
n1
∞
n
n
1 1X
1 X
1 X
1 X
≤ +
|qi xk (i)| + p
|qi xk (i)| + δ
2 2 n=1 Qn i=1
2 n=n +1 Qn i=1
1
!p
!p
∞
∞
n
n
p
X
X
1 1
1
1 X
2 −1 X
≤ +
|qi xk (i)| −
|qi xk (i)| + δ
2 2 n=1 Qn i=1
2p n=n +1 Qn i=1
n1
1X
≤
2 n=1
1
< 1 − 2δ + δ = 1 − δ.
That is a contradiction. Hence (2.1.1) must hold. Since
!p
!p
n1
n1
n
X
1 X
1 X
|qi xk (i)|
≤
|qi xk (i)|
≤ 1,
Qn1 i=1
Qn i=1
n=1
we have |qi xk (i)| ≤ Qn1 for k ∈ N and i = 1, 2, . . . , n1 . Hence there is a subsequence (zn ) of (xn ) and a sequence (an ) of real numbers such that
lim qi zk (i) = ai ,
k→∞
for ı = 1, 2, . . . , n1 .
Therefore,
n1
X
n
1 X
|qi zk (i) − qi zm (i)|
Q n i=1
n=1
!p
< δ for n,m sufficiently large.
Consequently,
kzk − zm kp
∞
n
X
1 X
(
|qi zk (i) − qi zm (i)|)p
Q
n
n=1
i=1
!p
!p
n1
n
∞
n
X
X
1 X
1 X
=
|qi zk (i) − qi zm (i)| +
|qi zk (i) − qi zm (i)|
Qn i=1
Qn i=1
n=1
n=n1 +1
!p
n1
n
X
1 X
≤
|qi zk (i) − qi zm (i)|
Qn i=1
n=1
!p
!p !
∞
n
∞
n
X
X
1 X
1 X
p
+2
(
|qi zk (i)| +
|qi zm (i)|
Qn i=1
Qn i=1
n=n +1
n=n +1
=
1
≤δ+2
p+1
1
δ < ε0 ,
1/p
i.e., sep(xn ) ≤ sep(zn ) < ε0 . This is a contradiction. Therefore Cesp (q) must
satisfy the property (β).
5
GENERALIZED CESÀRO SEQUENCE SPACE
Theorem 2.2. C(Cesp (q)) = 21/p .
Proof. Let
in
X
K = sup{A(un ) : un =
un (i)ei ∈ S(Cesp (q)),
i=in−1 +1
ω
0 = i0 < i1 < i2 < . . . , un →0, .
Then C(Cesp (q)) ≥ K. Moreover, for any ε > 0, there is a sequence (xn ) ⊂
ω
S(Cesp (q)) with xn →0 such that
A(xn ) + ε > C(Cesp (q)).
By the definition of A(xn ), there exists a subsequence (yn ) of (xn ) such that
kyn + ym k + 2ε > C(Cesp (q))
(2.2.1)
for any n, m ∈ N with m 6= n. Take v1 = y1 . Then, by the absolute continuity of
the norm of y1 , there exists i1 ∈ N such that
∞
X
v1 (i)ei < ε.
i=i1 +1
Putting z1 =
i1
P
v1 (i)ei , we have
i=1
kz1 + ym k = ky1 + ym −
∞
X
v1 (i)ei k ≥ ky1 + ym k − ε
for any m > 1.
i=i1 +1
Hence by (2.2.1), we have
kz1 + ym k + 3ε > C(Cesp (q))
for any m > 1.
Since yn (i) → 0 for i = 1, 2, · · · , there exists n2 ∈ N with n2 > n1 such that
i
1
X
yn (i)ei < ε
whenever n ≥ n2 .
i=1
Define v2 = yn2 . Then there is i2 > i1 such that
∞
X
v2 (i)ei < ε.
i=i2 +1
Taking z2 =
i2
P
v2 (i)ei , we obtain
i=i1 +1
kz1 + z2 k = ky1 −
∞
X
v1 (i)ei + yn2 −
i=i1 +1
≥ ky1 + yn2 k − 3ε.
i1
X
i=1
v2 (i)ei −
∞
X
i=i2 +1
v2 (i)ei
6
V. A. KHAN
Hence and by (2.2.1), we immediately obtain
kz1 + z2 k + 5ε > C(Cesp (q)).
k−1
Suppose that increasing sequences (ij )k−1
j=1 , (nj )j=1 of natural numbers and a sek−1
quence (zj )j=1 of elements of Cesp (q) are already defined and
kzn + zm k + 6ε > C(Cesp (q))
for m, n ∈ {1, 2, . . . , k − 1}, m 6= n.
Since yn (i) → 0 for i = 1, 2, · · · , there exists a natural number nk > nk−1 such
that
i
k−1
X
yn (i)ei < ε
i=1
provided n ≥ nk . Put v = ynk . Then there is ik > ik−1 such that
∞
X
vk (i)ei < ε.
i=ik +1
Defining zk =
ik
P
vk (i)ei , we obtain
i=ik−1 +1
ij−1
ik−1
∞
∞
X
X
X
X
kzj + zk k = ynj −
vj (i)ei −
vj (i)ei + ynk −
vk (i)ei −
vk (i)ei i=1
ij+1
i=1
ik+1
≥ kynj + ynk k − 4ε
for j = 1, 2, . . . , k − 1.
Hence, by (2.2.1), we obtain
kzj + zk k + 6ε > C(Cesp (q))
for j = 1, 2, · · · , k − 1.
Using the induction principle, we can find a sequence (zn ) satisfying the following
conditions:
P
(1) zn =
vn (i)ei , where 0 = i0 < i1 < i2 < . . . ;
i=in−1 +1
(2) kzn + zm k + 6ε > C(Cesp (q)) for m, n, ∈ N, m 6= n;
(3) kzn k ≤ 1 for n = 1, 2, . . .;
ω
(4) zn →0 as n → ∞.
Define un = zn /kzn k for each n ∈ N. Then every un ∈ S(Cesp (q)) and
zn
zm ≥ kzn + zm k ≥ C(Cesp (q)) − 6ε
kun + um k = +
kzn k kzm k for any m, n ∈ N, m 6= n. By the arbitrariness of ε, we have C(Cesp (q)) = K.
Let ε > 0 be given. Take nε ∈ N such that
p
∞
X
a
< ε,
Qk
k=inε +1
7
GENERALIZED CESÀRO SEQUENCE SPACE
where
inε
X
a=
|qi unε (i)|.
i=inε −1 +1
Hence for any m > nε , we have
!p
k
∞
X
X
1
+ um k p =
|qi unε (i)| +
Qk i=1
i=inε −1 +1
k=im−1 +1
!p
im−1
k
∞
X
X
1 X
|qi unε (i)| +
≥
Qk i=1
i=inε −1 +1
k=im−1 +1
!p
k
∞
X
1 X
=
|qi unε (i)|
Qk i=1
i=i
+1
im−1
kunε
X
!p
k
X
1
(a +
|qi um (i)|
Qk
i=1
!p
k
1 X
(
|qi um (i)|
Qk i=1
nε −1
p
∞
X
a
−
+
Qk
k=im−1
∞
X
k=im−1 +1
!p
k
1 X
(
|qi um (i)|
Qk i=1
> 1 − ε + 1 = 2 − ε,
i. e. A(un ) ≥ (2 − ε)) ≥ (2 − ε)1/p .
On the other hand, for ε mentioned above, by Lemma 1.1, there exists δ > 0
such that
k|x + ykp − kxkp | < ε
whenever kxkp ≤ 1 and kykp < δ. Take nδ ∈ N such that
∞
X
k=inδ +1
a p
(
) < δ,
Qk
and
inδ
X
a=
|qi unδ (i)|.
i=inδ −1 +1
Hence for any m > nδ , we have
kunδ + um kp
im−1
X
=
i=inδ −1 +1
∞
X
≤
i=inδ −1 +1
k
1 X
|qi unδ (i)|
Qk i=1
1
Qk
= kunδ (i)kp +
k
X
−
k1=im−1 +1
< 2 + ε,
+
|qi unδ (i)|
i=1
∞
X
∞
X
k=im−1 +1
!p
k=im−1 +1
∞
X
!p
+
∞
X
k=im−1 +1
k
X
1
(a +
|qi um (i)|)
Qk
i=1
!p
k
1 X
|qi um (i)|
Qk i=1
+ kum kp
k
X
1
(a +
|qi um (i)|)
Qk
i=1
!p
k
X
1
(a +
|qi um (i)|)
Qk
i=1
!p
!p
8
V. A. KHAN
i. e. A((un )) ≤ (2 + ε)1/p .
Since ε was arbitrary, we obtain C(Cesp (q)) = 21/p .
Corollary 2.1. The space Cesp (q)) satisfies the Banach-Saks property .
Proof. The proof follows immediately from the above Theorem 1 of [3] and
Theorem 2.2.
Acknowledgment. The author is grateful to Prof. Mursaleen and to the referee for their valuable suggestions.
References
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51–57.
2. Clarkson J. A., Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
3. Cui Y. A. and Meng C. and Pluciennik, R., Banach-Saks property and property (β) in
Cesàro Sequence Space, Southeast Asian Bull. Math. 24 (2000), 201–210.
4. Cui Y. A. and Hudzik H., On the Banach-Saks and weak Banach-Saks properties of some
Banach sequence spaces, Acta Sci. Math. (Szeged), 65 (1999), 179–187.
5. Diestel J., Sequences and series in Banach spaces, Graduate Texts in Math. 92, Springer-Verlag, 1984.
6. Kakutani S., Weak convergence in Uniformly convex Banach spaces, Tohoku Math. J. Math.
45 (1938), 188–193.
7. Kotmann C. A., Packing and reflexivity in Banach Spaces, Trans. Amer. Math. Soc. 150
(1970), 565–576.
8. Lee. P. Y., Cesàro sequence Spaces, Math cronical, New Zeland 13 (1984), 29–45.
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V. A. Khan, Department of Mathematics, A. M. U. Aligarh-202002, India,
e-mail: vakhan@math.com