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Some geometric properties of -spaces
Journal of Inequalities and Applications volume 2014, Article number: 112 (2014)
Abstract
In this paper, we introduce the sequence spaces and we show some geometric properties of that spaces. The main purpose of this paper is to show that is a Banach space and has the rotund property, the Kadec-Klee property, the uniform Opial property, the -property, the k-NUC property and the Banach-Saks property of type p.
MSC:46A45, 40C05, 46B45, 46A35.
1 Introduction
By ω, we denote the space of all real valued sequences. Any vector subspace of ω is called a sequence space. We write , c, and for the spaces of all bounded, convergent and null sequences, respectively. Also by bs, cs, , and , we denote the spaces of all bounded, convergent, absolutely convergent and p-absolutely convergent series, respectively; where . Assume here and after that be a bounded sequence of strictly positive real numbers with and . Then, the linear space was defined by Maddox [1] (see also Simons [2] and Nakano [3]) as follows:
which is complete paranormed space paranormed by
For simplicity in notation, here and in what follows, the summation without limits runs from 1 to ∞.
In [4] was introduced the following numerical sequence , which is a strictly increasing sequence of positive real numbers tending to infinity, as , that is
We will introduce the following sequence space:
For , we obtain the Cesaro sequence space (see [5]). If and , then (see [6]). In case where for all , then we will denote . Some results related to the geometric properties of sequence spaces are given in [7–9].
2 Topological properties
Theorem 2.1 The paranorm on is given by the relation
where and .
3 Geometrical properties
In this section we will show some geometric properties of the -spaces, such as the -property, the k-NUC property, the Banach-Saks property of type p, and the -property. It is well known that these properties are most important in Banach spaces (see [10, 11] and [1]).
Definition 3.1 A Banach space X is said to be k-nearly uniformly convex (k-NUC) if for any , there exists a such that for any sequence with , there are such that
where .
Definition 3.2 A Banach space X has property if and only if for each and , there exists a such that for each element and each sequence with , there is an index k for each
Definition 3.3 A Banach space X is said to have the Banach-Saks property type p if every weakly null sequence has a subsequence such that for some
for all .
Definition 3.4 Let X be a real vector space. A functional is called a modular if
-
(1)
if and only if ,
-
(2)
for all scalars α with ,
-
(3)
for all and with ,
-
(4)
the modular σ is called convex if for all and , with .
A modular σ is called:
-
(5)
right continuous if for all ,
-
(6)
left continuous if for all ,
-
(7)
continuous if it is both right and left continuous,
where . We define on as follows:
where .
If , for all , by the convexity of the function , for all , defined above is a modular convex in the .
Definition 3.5 A modular is said to satisfy the -conditions if for every , there exist constant and such that
for all with .
Lemma 3.6 ([12])
If satisfies the -conditions, then for any and , there exists such that
whenever with and .
Theorem 3.7 ([12])
-
(1)
If satisfies the -conditions, then for any , if and only if .
-
(2)
If satisfies the -conditions, then for any sequence , if and only if .
Theorem 3.8 If satisfies the -conditions, then for any , there exists such that implies .
Proof The proof of the theorem follows directly from the above two facts. □
Theorem 3.9 For any and , there exists , such that implies .
Proof The proof of the theorem follows directly from Theorem 3.8. □
Proposition 3.10 If , for all , then the modular function , on , satisfies the following conditions:
-
(1)
If , then and .
-
(2)
If , then .
-
(3)
If , then .
-
(4)
The modular function is continuous on .
Proof The proof of the proposition is similar to Proposition 2.1 in [13]. □
Now we will define the following two norms (the first is known as the Luxemburg norm and the second as the Amemiya norm) in :
and
Proposition 3.11 Let . Then the following relations are satisfied:
-
(1)
If , then .
-
(2)
If , then .
-
(3)
if and only if .
-
(4)
if and only if .
-
(5)
if and only if .
Proof (1) Let and . Let also such that . On the other hand from the definition of the norm by relation (3.3) we find that there exists a such that and . From the above relations and property (1) of Proposition 3.10, we obtain
and
The previous statement is valid for every , from which it follows that .
-
(2)
In this case we will choose such that , and we obtain . Now using into consideration definition of the norm (3.3) and relation (1) of Proposition 3.10, we get
for every . Finally we have proved that .
-
(3)
Since is continuous function (see [14]), this property follows immediately.
-
(4)
Follows from properties (1) and (3).
-
(5)
Follows from properties (2) and (3). □
Theorem 3.12 is a Banach space under the Luxemburg and Amemiya norms.
Proof We will prove that is a Banach space under the Luxemburg norm. In a similar way we can prove that is a Banach space under the Amemiya norm. In what follows we need to show that every Cauchy sequence in is convergent according to the Luxemburg norm. Let be any Cauchy sequence in and . Thus there exists a natural number , such that for any we get . From Proposition 3.11 we get
for all . This implies that
For each fixed k and for all ,
Hence is a Cauchy sequence in ℝ. Since ℝ is a complete normed space, there exists a such that as . Therefore, as , by relation (3.6) we have
for all . In the sequel we will show that is a sequence form . From Proposition 3.10 and relation (3.5) we have
for all . This implies that as . So we have . This proves that is a complete normed space under the Luxemburg norm. □
In what follows we will show results related to the Luxemburg norm, and for this reason we will denote it just .
Theorem 3.13 The space is rotund if and only if for all .
Proof Let be rotund and choose such that . Consider the two sequences given by
and
Then obviously and
Then from Proposition 3.11, property (3), it follows that , which leads to the contradiction that the sequence space is not rotund. Hence , for all .
Conversely, let and such that . By the convexity of and property (3) from Proposition 3.11, we have
which gives and
From the previous relation we obtain
Since , we get
This implies that
From the previous relation we get for all , respectively, . That is, the sequence space is rotund. □
In what follows we will give two facts without proof because their proofs follow directly from Proposition 3.10 and Proposition 3.11.
Theorem 3.14 Let . Then the following statements hold:
-
(i)
For and we get .
-
(ii)
If and , then we have .
Theorem 3.15 Let be a sequence in . Then the following statements hold:
-
(i)
implies .
-
(ii)
implies .
Theorem 3.16 Let and . If as and as for all , then as .
Proof The proof of the theorem is similar to Theorem 2.9 in [13]. □
Theorem 3.17 The sequence space has the Kadec-Klee property.
Proof It is enough to prove that every weakly convergent sequence on the unit sphere is convergent in norm. Let and such that and be given. From the properties of Theorem 3.15 it follows that as . On the other hand, from Proposition 3.11, we get . Therefore we have , as . Since and is a continuous functional, as and for . Now the proof of the theorem follows from Theorem 3.16. □
Theorem 3.18 For any , the space has the uniform Opial property.
We omit this proof.
To prove the following theorem we will use the same technique given in [15] and will consider that .
Theorem 3.19 The Banach space has the k-NUC property for every .
Proof Let and with . For each , let
Since for each , is bounded, by the diagonal method (see [16]), we find that for each , we can find a subsequence of such that converges for each , . Therefore, there exists an increasing sequence of positive integers such that . Hence, there is a sequence of positive integers with such that for all . Then by Theorem 3.15, we may assume that there exists such that
Let be such that . For fixed integer , let . Then by Lemma 3.6, there is a such that
whenever and . Since by Proposition 3.11, property (1), we get , . Then there exist positive integers () with such that and for all . Define . By (3.10), we have . Let for and . From relations (3.10), (3.11), and the convexity of the function (), we have
from (3.11) we get
from the convexity of (), it follows that
Now from Theorem 3.9, there exists a such that
□
The proof of the following results we omit.
Theorem 3.20 The Banach space has the -property.
Theorem 3.21 The Banach space has the Banach-Saks property of type p.
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Braha, N.L. Some geometric properties of -spaces. J Inequal Appl 2014, 112 (2014). https://doi.org/10.1186/1029-242X-2014-112
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DOI: https://doi.org/10.1186/1029-242X-2014-112