On some imbedding relations between certain sequence spaces
In the present paper, we introduce the sequence space $l^{\lambda}_p$ of non-absolute type which is a $p$-normed space and a $BK$-space in the cases of $0 < p < 1$ and $0 < p < 1$ i $1 \leq p < \infty$, respectively. Further, we derive some imbedding relations and...
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Institute of Mathematics, NAS of Ukraine
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| author | Mursaleen, M. Noman, A. K. Мурзаліїв, М. Номан, А. К. |
| author_facet | Mursaleen, M. Noman, A. K. Мурзаліїв, М. Номан, А. К. |
| author_sort | Mursaleen, M. |
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| description | In the present paper, we introduce the sequence space $l^{\lambda}_p$ of non-absolute type which is a $p$-normed space and a $BK$-space in the cases of $0 < p < 1$ and $0 < p < 1$ i $1 \leq p < \infty$, respectively. Further, we derive some imbedding relations and construct the basis for the space $l^{\lambda}_p$, where $1 \leq p < \infty$. |
| first_indexed | 2026-03-24T02:29:17Z |
| format | Article |
| fulltext |
UDC 517.9
M. Mursaleen, A. K. Noman (Aligarh Muslim Univ., India)
ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN
SEQUENCE SPACES*
ПРО ДЕЯКI СПIВВIДНОШЕННЯ ВКЛАДЕННЯ
МIЖ ПЕВНИМИ ПРОСТОРАМИ ПОСЛIДОВНОСТЕЙ
In the present paper, we introduce the sequence space `λp of non-absolute type which is a p-normed space and
a BK-space in the cases of 0 < p < 1 and 1 ≤ p < ∞, respectively. Further, we derive some imbedding
relations and construct the basis for the space `λp , where 1 ≤ p <∞.
Введено поняття простору послiдовностей `λp неабсолютного типу, який є p-нормованим простором i
BK-простором у випадках 0 < p < 1 i 1 ≤ p < ∞ вiдповiдно. Крiм того, отримано деякi спiввiдно-
шення вкладення та побудовано базис для простору `λp , де 1 ≤ p <∞.
1. Introduction. By w, we denote the space of all complex valued sequences. Any
vector subspace of w is called a sequence space.
A sequence space E with a linear topology is called a K-space provided each of the
maps pi : E → C defined by pi(x) = xi is continuous for all i ∈ N; where C denotes the
complex field and N = {0, 1, 2, . . .}. A K-space E is called an FK-space provided E
is a complete linear metric space. An FK-space whose topology is normable is called a
BK-space [2, p. 1451], that is, a BK-space is a Banach sequence space with continuous
coordinates [11, p. 187].
We shall write `∞, c and c0 for the sequence spaces of all bounded, convergent and
null sequences, respectively, which are BK-spaces with the usual sup-norm defined by
‖x‖`∞ = sup
k
|xk|,
where, here and in the sequel, the supremum supk is taken over all k ∈ N. Also, by `p,
0 < p < ∞, we denote the sequence space of all p-absolutely convergent series. It is
known that the space `p is a complete p-normed space and a BK-space in the cases of
0 < p < 1 and 1 ≤ p <∞, respectively, with respect to the usual p-norm and `p-norm
defined by
‖x‖`p =
∑
k
|xk|p, 0 < p < 1,
and
‖x‖`p =
(∑
k
|xk|p
)1/p
, 1 ≤ p <∞,
respectively. For simplicity in notation, here and in what follows, the summation without
limits runs from 0 to ∞.
Let X and Y be sequence spaces and A = (ank) be an infinite matrix of complex
numbers ank, where n, k ∈ N. Then, we say that A defines a matrix mapping from X
into Y, and we denote it by writing A : X → Y, if for every sequence x = (xk) ∈ X
the sequence Ax = {An(x)}, the A-transform of x, exists and is in Y, where
*Research of the second author was supported by Department of Mathematics, Faculty of Education and
Science, Al Baida University, Yemen.
c© M. MURSALEEN, A. K. NOMAN, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4 489
490 M. MURSALEEN, A. K. NOMAN
An(x) =
∑
k
ankxk, n ∈ N. (1.1)
By (X : Y ), we denote the class of all infinite matrices A = (ank) such that A : X →
→ Y. Thus,A ∈ (X : Y ) if and only if the series on the right-hand side of (1.1) converges
for each n ∈ N and every x ∈ X, and Ax ∈ Y for all x ∈ X. A sequence x is said to
be A-summable to l ∈ C if Ax coverges to l which is called the A-limit of x.
For a sequence space X, the matrix domain of an infinite matrix A in X is defined
by
XA =
{
x ∈ w : Ax ∈ X
}
(1.2)
which is a sequence space.
We shall write e(k) for the sequence whose only non-zero term is a 1 in the k th
place for each k ∈ N.
The approach constructing a new sequence space by means of the matrix domain
of a particular limitation method has recently been employed by several authors in
many research papers (see, for example, [1 – 7,12 – 15,17, 18]). The main purpose of this
paper is to introduce the sequence space `λp of non-absolute type and is to derive some
related results. Further, we establish some imbedding relations concerning the space `λp ,
0 < p <∞. Finally, we construct the basis for the space `λp , where 1 ≤ p <∞.
2. The sequence space `λp of non-absolute type. Throughout this paper, let λ =
= (λk)k∈N be a strictly increasing sequence of positive reals tending to ∞, that is
0 < λ0 < λ1 < . . . and λk →∞ as k →∞. (2.1)
By using the convention that any term with a negative subscript is equal to naught,
we define the infinite matrix Λ = (λnk) by
λnk =
λk − λk−1
λn
, 0 ≤ k ≤ n,
0, k > n,
(2.2)
for all n, k ∈ N. Then, it is obvious by (2.2) that the matrix Λ = (λnk) is a triangle,
that is λnn 6= 0 and λnk = 0 for all k > n, n ∈ N. Further, by using (1.1), we have for
every x = (xk) ∈ w that
Λn(x) =
1
λn
n∑
k=0
(λk − λk−1)xk, n ∈ N. (2.3)
Recently, Mursaleen and Noman [14] introduced the sequence spaces cλ0 , c
λ and `λ∞
as follows:
cλ0 =
{
x ∈ w : lim
n
Λn(x) = 0
}
,
cλ =
{
x ∈ w : lim
n
Λn(x) exists
}
and
`λ∞ =
{
x ∈ w : sup
n
|Λn(x)| <∞
}
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN SEQUENCE SPACES 491
Moreover, it has been shown that the inclusions c0 ⊂ cλ0 , c ⊂ cλ and `∞ ⊂ `λ∞ hold.
We refer the reader to [14] for relevant terminology.
Now, as a natural continuation of the above spaces, we define `λp as the set of all
sequences whose Λ-transforms are in the space `p, 0 < p <∞; that is
`λp =
{
x ∈ w :
∑
n
|Λn(x)|p <∞
}
, 0 < p <∞.
With the notation of (1.2), we may redefine the space `λp , 0 < p <∞ as the matrix
domain of the triangle Λ in the space `p. This can be written as follows:
`λp = (`p)Λ, 0 < p <∞. (2.4)
It is trivial that `λp , 0 < p < ∞, is a linear space with the coordinatewise addition
and scalar multiplication. Further, it follows by (2.4) that the space `λp , 0 < p < 1,
becomes a p-normed space with the following p-norm:
‖x‖`λp = ‖Λ(x)‖`p =
∑
n
|Λn(x)|p, 0 < p < 1.
Moreover, since the matrix Λ is a triangle, we have the following result which is
essential in the text.
Theorem 2.1. The sequence space `λp , 1 ≤ p <∞, is a BK-space with the norm
given by
‖x‖`λp = ‖Λ(x)‖`p =
(∑
n
|Λn(x)|p
)1/p
, 1 ≤ p <∞. (2.5)
Proof. Since (2.4) holds and `p, 1 ≤ p < ∞, is a BK-space with the `p-norm (see
[10, p. 218]), this result is immediate by Theorem 4.3.12 of Wilansky [19, p. 63].
Remark 2.1. One can easily check that the absolute property does not hold on the
space `λp , 0 < p < ∞, that is ‖x‖`λp 6= ‖|x|‖`λp for at least one sequence x ∈ `λp . This
tells us that `λp is a sequence space of non-absolute type, where |x| = (|xk|) .
Theorem 2.2. The sequence space `λp of non-absolute type is linear isometric to
the space `p, where 0 < p <∞.
Proof. To prove this, we should show the existence of a linear isometry between
the spaces `λp and `p, where 0 < p < ∞. For this, let us consider the transformation
T defined, with the notation of (2.3), from `λp to `p by x 7−→ Λ(x) = Tx. Then
Tx = Λ(x) ∈ `p for every x ∈ `λp . Also, the linearity of T is trivial. Further, it is easiy
to see that x = 0 whenever Tx = 0 and hence T is injective.
Furthermore, for any given y = (yk) ∈ `p, we define the sequence x = (xk) by
xk =
λkyk − λk−1yk−1
λk − λk−1
, k ∈ N.
Then, we have for every n ∈ N that
Λn(x) =
1
λn
n∑
k=0
(λk − λk−1)xk =
1
λn
n∑
k=0
(λkyk − λk−1yk−1) = yn.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
492 M. MURSALEEN, A. K. NOMAN
This shows that Λ(x) = y and since y ∈ `p, we obtain that Λ(x) ∈ `p. Thus, we deduce
that x ∈ `λp and Tx = y. Hence, the operator T is surjective.
Moreover, let x ∈ `λp be given. Then, we have that
‖Tx‖`p = ‖Λ(x)‖`p = ‖x‖`λp
and hence T is an isometry. Consequently, the spaces `λp and `p are linear isometric for
0 < p <∞.
Theorem 2.2 is proved.
Finally, we know that the space `2 is the only Hilbert space among the Banach spaces
`p, 1 ≤ p < ∞. Thus, we conclude this section with the following corollary which is
immediate by Theorems 2.1 and 2.2.
Corollary 2.1. Except the case p = 2, the space `λp is not an inner product space,
hence not a Hilbert space for 1 ≤ p <∞.
3. Some imbedding relations. In the present section, we establish some imbedding
relations concerning the space `λp , 0 < p < ∞. We essentially characterize the case in
which the imbedding `p ⊂ `λp holds for 1 ≤ p <∞.
The notion of imbedded Banach spaces can be found in [9] (Chapter I) and it can be
given as follows:
Let X and Y be Banach spaces. Then, we say that X is imbedded in Y if the
following conditions are satisfied:
(i) x ∈ X implies x ∈ Y, that is, the space Y includes X.
(ii) The space Y includes a vector space structure on X coinciding with the structure
of X.
(iii) There exists a constant C > 0 such that ‖x‖Y ≤ C ‖x‖X for all x ∈ X.
In what follows, we shall denote the imbedding of X in Y by X ⊂ Y, assuming
that the symbol ⊂ means not only the set-theoretic inclusion, but imbedding have the
properties (ii) and (iii). Further, we say that the imbedding X ⊂ Y strictly holds if the
space Y strictly includes X.
Since any two sequence spaces have the same vector space structure, the condition
(ii) is redundant when X and Y are BK-spaces.
Now, we may begin with the following basic result:
Theorem 3.1. If 0 < p < s <∞, then the imbedding `λp ⊂ `λs strictly holds.
Proof. Since the space `s strictly includes `p, the space `λs strictly includes `λp .
Therefore, this result is immediate by the fact that the topology of the space `λp is
stronger than the topology of `λs , that is
‖x‖`λs = ‖Λ(x)‖`s ≤ ‖Λ(x)‖`p = ‖x‖`λp
for all x ∈ `λp , where 0 < p < s <∞.
Theorem 3.1 is proved.
Although the imbeddings c0 ⊂ cλ0 , c ⊂ cλ and `∞ ⊂ `λ∞ always holds, the space `p
may not be included in `λp for 0 < p < ∞. This will be shown in the following lemma
in which we write
1
λ
=
(
1
λk
)
.
Lemma 3.1. Let 0 < p < ∞. Then, the spaces `p and `λp overlap. Further, if
1
λ
/∈ `p then neither of the spaces `p and `λp includes the other one.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN SEQUENCE SPACES 493
Proof. Obviously, the spaces `p and `λp always overlap, since the sequence (λ1 −
− λ0,−λ0, 0, 0, . . .) belongs to both spaces `p and `λp for 0 < p <∞.
Suppose now that
1
λ
/∈ `p, 0 < p < ∞, and consider the sequence x = e(0) =
= (1, 0, 0, . . .) ∈ `p. Then, by using (2.3), we have for every n ∈ N that
Λn(x) =
1
λn
n∑
k=0
(λk − λk−1)e
(0)
k =
λ0
λn
.
Thus, we obtain that ∑
n
|Λn(x)|p = λp0
∑
n
1
λpn
which shows that Λ(x) /∈ `p and hence x /∈ `λp . Thus, the sequence x is in `p but not in
`λp . Hence, the space `λp does not include `p when
1
λ
/∈ `p, where 0 < p <∞.
On the other hand, let 1 ≤ p <∞ and define the sequence y = (yk) by
yk =
1
λk
, k is even,
k ∈ N.
− 1
λk−1
(λk−1 − λk−2
λk − λk−1
)
, k is odd,
Then y /∈ `p, since
1
λ
/∈ `p. Besides, we have for every n ∈ N that
Λn(y) =
1
λn
(λn − λn−1
λn
)
, n is even,
0, n is odd.
Thus, we obtain that∑
n
|Λn(y)|p =
∑
n
|Λ2n(y)|p =
∑
n
1
λp2n
(
λ2n − λ2n−1
λ2n
)p
≤
≤ 1
λp0
+
∞∑
n=1
1
λp2n−2
(
λ2n − λ2n−2
λ2n
)p
≤
≤ 1
λp0
+
∞∑
n=1
1
λp2n−2
(
λp2n − λ
p
2n−2
λp2n
)
=
=
1
λp0
+
∞∑
n=1
( 1
λp2n−2
− 1
λp2n
)
=
2
λp0
<∞.
This shows that Λ(y) ∈ `p and hence y ∈ `λp . Thus, the sequence y is in `λp but not in
`p, where 1 ≤ p <∞.
Similarly, one can construct a sequence belonging to the set `λp \ `p for 0 < p < 1.
Therefore, the space `p also does not include `λp when
1
λ
/∈ `p for 0 < p <∞.
Lemma 3.1 is proved.
As an immediate consequence of Lemma 3.1, we have the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
494 M. MURSALEEN, A. K. NOMAN
Lemma 3.2. If the imbedding `p ⊂ `λp holds, then
1
λ
∈ `p, where 0 < p <∞.
Proof. Suppose that the imbedding `p ⊂ `λp holds, where 0 < p < ∞, and consider
the sequence x = e(0) = (1, 0, 0, . . .) ∈ `p. Then x ∈ `λp and hence Λ(x) ∈ `p. Thus,
we obtain that
λp0
∑
n
(
1
λn
)p
=
∑
n
|Λn(x)|p <∞
which shows that
1
λ
∈ `p.
Lemma 3.2 is proved.
We shall later show that the condition
1
λ
∈ `p is not only necessary but also sufficient
for the imbedding `p ⊂ `λp to be held, where 1 ≤ p < ∞. Before that, by taking into
account the definition of the sequence λ = (λk) given by (2.1), we find that
0 <
λk − λk−1
λn
< 1, 0 ≤ k ≤ n,
for all n, k ∈ N with n + k > 0. Furthermore, if
1
λ
∈ `1 then we have the following
lemma which is easy to prove.
Lemma 3.3. If
1
λ
∈ `1, then
sup
k
(λk − λk−1)
∞∑
n=k
1
λn
<∞.
Now, we prove the following:
Theorem 3.2. The imbedding `1 ⊂ `λ1 holds if and only if
1
λ
∈ `1.
Proof. The necessity is immediate by Lemma 3.2.
Conversely, suppose that
1
λ
∈ `1. Then, it follows by Lemma 3.3 that
M = sup
k
(λk − λk−1)
∞∑
n=k
1
λn
<∞.
Therefore, we have for every x = (xk) ∈ `1 that
‖x‖`λ1 =
∑
n
|Λn(x)| ≤
∞∑
n=0
1
λn
n∑
k=0
(λk − λk−1)|xk| =
=
∞∑
k=0
|xk|(λk − λk−1)
∞∑
n=k
1
λn
≤M
∞∑
k=0
|xk| = M‖x‖`1 .
This also shows that the space `λ1 includes `1. Hence, the imbedding `1 ⊂ `λ1 holds
which concludes the proof.
Corollary 3.1. If
1
λ
∈ `1, then the imbedding `p ⊂ `λp holds for 1 ≤ p <∞.
Proof. The imbedding trivially holds for p = 1 by Theorem 3.2, above. Thus, let
1 < p < ∞ and take any x ∈ `p. Then |x|p ∈ `1 and hence |x|p ∈ `λ1 by Theorem 3.2
which implies that x ∈ `λp . This shows that the space `p is included in `λp .
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN SEQUENCE SPACES 495
Further, let x = (xk) ∈ `p be given. Then, for every n ∈ N, we obtain by applying
the Hölder’s inequality that
|Λn(x)|p ≤
[
n∑
k=0
(
λk − λk−1
λn
)
|xk|
]p
≤
≤
[
n∑
k=0
(
λk − λk−1
λn
)
|xk|p
][
n∑
k=0
λk − λk−1
λn
]p−1
=
1
λn
n∑
k=0
(λk − λk−1)|xk|p.
Thus, we derive that
‖x‖p
`λp
=
∑
n
|Λn(x)|p ≤
∞∑
n=0
1
λn
n∑
k=0
(λk − λk−1)|xk|p =
=
∞∑
k=0
|xk|p(λk − λk−1)
∞∑
n=k
1
λn
≤M
∞∑
k=0
|xk|p = M‖x‖p`p ,
whereM = supk
[
(λk − λk−1)
∑∞
n=k
1
λn
]
<∞ by Lemma 3.3. Hence, the imbedding
`p ⊂ `λp also holds for 1 < p <∞.
Corollary 3.1 is proved.
Now, as a generalization of Theorem 3.2, the following theorem shows the necessity
and sufficiency of the condition
1
λ
∈ `p for the imbedding `p ⊂ `λp to be held, where
1 ≤ p <∞.
Theorem 3.3. The imbedding `p ⊂ `λp holds if and only if
1
λ
∈ `p, where 1 ≤ p <
<∞.
Proof. The necessity is trivial by Lemma 3.2. Thus, we turn to the sufficiency. For
this, suppose that
1
λ
∈ `p, where 1 ≤ p < ∞. Then
1
λp
=
(
1
λpk
)
∈ `1. Therefore, it
follows by Lemma 3.3 that
sup
k
(λk − λk−1)p
∞∑
n=k
1
λpn
≤ sup
k
(λpk − λ
p
k−1)
∞∑
n=k
1
λpn
<∞.
Further, we have for every fixed k ∈ N that
Λn
(
e(k)
)
=
λk − λk−1
λn
, 0 ≤ k ≤ n,
n ∈ N.
0, k > n,
Thus, we obtain that
‖e(k)‖p
`λp
= (λk − λk−1)p
∞∑
n=k
1
λpn
<∞, k ∈ N,
which yields that e(k) ∈ `λp for every k ∈ N, i.e., every basis element of the space `p
is in `λp . This shows that the space `λp contains the Schauder basis for the space `p such
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
496 M. MURSALEEN, A. K. NOMAN
that
sup
k
‖e(k)‖`λp <∞.
Therefore, we deduce that the space `λp includes `p. Moreover, by using the same
technique used in the proof of Corollary 3.1, it can similarly be shown that the topology
of the space `p is stronger than the topology of `λp . Hence, the imbedding `p ⊂ `λp holds,
where 1 ≤ p <∞.
Theorem 3.3 is proved.
Now, in the following example, we give an important particular case of the space
`λp , where 1 ≤ p <∞.
Example 3.1. Consider the particular case of the sequence λ = (λk) given by
λk = k + 1 for all k ∈ N. Then
1
λ
/∈ `1 and hence `1 is not included in `λ1 by
Lemma 3.1.
On the other hand, we have
1
λ
∈ `p for 1 < p < ∞ and so `p is included in `λp .
Further, by applying the well-known inequality (see [8, p. 239])
∑
n
(
n∑
k=0
|xk|
n+ 1
)p
<
(
p
p− 1
)p∑
n
|xn|p, 1 < p <∞,
we immediately obtain that
‖x‖`λp <
p
p− 1
‖x‖`p , 1 < p <∞,
for all x ∈ `p. This shows that the imbedding `p ⊂ `λp holds for 1 < p <∞. Moreover,
this imbedding is strict. For example, the sequence y = {(−1)k}k∈N is not in `p but in
`λp , since
∑
n
|Λn(y)|p =
∑
n
∣∣∣∣∣ 1
n+ 1
n∑
k=0
(−1)k
∣∣∣∣∣
p
=
∑
n
1
(2n+ 1)
p <∞, 1 < p <∞.
Remark 3.1. In the special case λk = k + 1 (k ∈ N) given in Example 3.1, we
may note that the space `λp is reduced to the Cesàro sequence space Xp of non-absolute
type, where 1 ≤ p <∞ (see [16, 17]).
Now, let x = (xk) be a null sequence of positive reals, that is
xk > 0 for all k ∈ N and xk → 0 as k →∞.
Then, as is easy to see, for every positive integer m there is a subsequence {xkr}r∈N of
the sequence x such that
xkr = O
(
1
(r + 1)
m+1
)
.
Further, this subsequence can be chosen such that kr+1 − kr ≥ 2 for all r ∈ N.
In general, if x = (xk) is a sequence of positive reals such that lim inf xk = 0, then
there is a subsequence x′ = {xk′r}r∈N of the sequence x such that limr xk′r = 0. Thus
x′ is a null sequence of positive reals. Hence, as we have seen above, for every positive
integer m there is a subsequence {xkr}r∈N of the sequence x′ and hence of the sequence
x such that kr+1 − kr ≥ 2 for all r ∈ N and
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ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN SEQUENCE SPACES 497
xkr = O
(
1
(r + 1)
m+1
)
,
where kr = k′θ(r) and θ : N→ N is a suitable increasing function. Thus, we obtain that
(r + 1)xkr = O
(
1
(r + 1)
m
)
.
Now, let 0 < p < ∞. Then, we can choose a positive integer m such that mp > 1.
In this situation, the sequence {(r + 1)xkr}r∈N is in the space `p.
Obviously, we observe that the subsequence {xkr}r∈N depends on the positive integer
m which is, in turn, depending on p. Thus, our subsequence depends on p.
Hence, from the above discussion, we conclude the following result:
Lemma 3.4. Let x = (xk) be a sequence of positive reals such that lim inf xk = 0.
Then, for every positive number p ∈ (0,∞) there is a subsequence x(p) = {xkr}r∈N of
x, depending on p, such that kr+1 − kr ≥ 2 for all r ∈ N and
∑
r |(r + 1)xkr |p <∞.
Moreover, we have the following two lemmas (see [14]) which are needed in the
sequel.
Lemma 3.5. For any sequence x = (xk) ∈ w, the equalities
Sn(x) = xn − Λn(x), n ∈ N, (3.1)
and
Sn(x) =
λn−1
λn − λn−1
[
Λn(x)− Λn−1(x)
]
, n ∈ N, (3.2)
hold, where the sequence S(x) = {Sn(x)} is defined by
S0(x) = 0 and Sn(x) =
1
λn
n∑
k=1
λk−1(xk − xk−1) for n ≥ 1.
Lemma 3.6. For any sequence λ = (λk) satisfying (2.1), the sequence{
λk
λk − λk−1
}
k∈N
is bounded if and only if lim inf
λk+1
λk
> 1, and is unbounded if
and only if lim inf
λk+1
λk
= 1.
Now, we know by Theorem 3.3 that the imbedding `p ⊂ `λp holds whenever
1
λ
∈ `p,
1 ≤ p < ∞. More precisely, the following theorem gives the necessary and sufficient
conditions for this imbedding to be strict.
Theorem 3.4. Let 1 ≤ p <∞. Then, the imbedding `p ⊂ `λp strictly holds if and
only if
1
λ
∈ `p and lim inf
λn+1
λn
= 1.
Proof. Suppose that the imbedding `p ⊂ `λp is strict, where 1 ≤ p < ∞. Then, the
necessity of the first condition is immediate by Theorem 3.3. Further, since `λp strictly
includes `p, there is a sequence x ∈ `λp such that x /∈ `p, that is Λ(x) ∈ `p while x /∈ `p.
Thus, we obtain by (3.1) of Lemma 3.5 that S(x) = {Sn(x)} /∈ `p. Moreover, since
Λ(x) ∈ `p, we have
∑
n
|Λn(x)|p < ∞ and hence
∑
n |Λn(x) − Λn−1(x)|p < ∞
by applying the Minkowski’s inequality. This means that {Λn(x) − Λn−1(x)} ∈ `p.
Thus, by combining this with the fact that {Sn(x)} /∈ `p, it follows by (3.2) that the
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
498 M. MURSALEEN, A. K. NOMAN
sequence
{
λn−1
λn − λn−1
}
is unbounded and hence
{
λn
λn − λn−1
}
/∈ `∞. This leads us
with Lemma 3.6 to the necessity of the second condition.
Conversely, since
1
λ
∈ `p, we have by Theorem 3.3 that the imbedding `p ⊂ `λp
holds. Further, since lim inf
λk+1
λk
= 1, we obtain by Lemma 3.6 that
lim inf
λk − λk−1
λk
= 0.
Thus, it follows by Lemma 3.4 that there is a subsequence λ(p) = {λkr}r∈N of the
sequence λ = (λk), depending on p, such that kr+1 − kr ≥ 2 for all r ∈ N and
∑
r
∣∣∣∣(r + 1)
(λkr − λkr−1
λkr
)∣∣∣∣p <∞. (3.3)
Let us now define the sequence y = (yk) for every k ∈ N by
yk =
r + 1, k = kr,
−(r + 1)
(
λk−1 − λk−2
λk − λk−1
)
, k = kr + 1, r ∈ N,
0, otherwise.
Then, it is clear that y /∈ `p. Moreover, we have for every n ∈ N that
Λn(y) =
(r + 1)
(
λn − λn−1
λn
)
, n = kr,
r ∈ N.
0, n 6= kr,
This and (3.3) imply that Λ(y) ∈ `p and hence y ∈ `λp . Thus, the sequence y is in `λp
but not in `p. Therefore, the imbedding `p ⊂ `λp strictly holds, where 1 ≤ p <∞.
Theorem 3.4 is proved.
As an immediate consequence of Theorem 3.4, we have the following result:
Theorem 3.5. The equality `λp = `p holds if and only if lim inf
λn+1
λn
> 1, where
1 ≤ p <∞.
Proof. The necessity is immediate by Theorems 3.3 and 3.4. For, if the equality
holds then `p is imbedded in `λp and hence
1
λ
∈ `p by Theorem 3.3. Further, since the
imbedding `p ⊂ `λp cannot be strict, we have by Theorem 3.4 that lim inf
λn+1
λn
6= 1 and
hence lim inf
λn+1
λn
> 1.
Conversely, suppose that lim inf
λn+1
λn
> 1. Then, there exists a constant a > 1 such
that
λn+1
λn
≥ a and hence λn ≥ λ0a
n for all n ∈ N. This shows that
1
λ
∈ `1 which
leads us with Corollary 3.1 to the consequence that the imbedding `p ⊂ `λp holds and
hence `p is included in `λp , where 1 ≤ p <∞.
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ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN SEQUENCE SPACES 499
On the other hand, by using Lemma 3.6, we have the bounded sequence{
λn
λn − λn−1
}
and hence
{
λn−1
λn − λn−1
}
∈ `∞.
Now, let x ∈ `λp . Then Λ(x) = {Λn(x)} ∈ `p and hence {Λn(x)− Λn−1(x)} ∈ `p.
Thus, we obtain by (3.2) that S(x) = {Sn(x)} ∈ `p. Therefore, it follows by (3.1) that
x = S(x) + Λ(x) ∈ `p. This shows that x ∈ `p for all x ∈ `λp and hence `λp is also
included in `p. Consequently, the equality `λp = `p holds, where 1 ≤ p <∞.
Theorem 3.5 is proved.
Finally, we conclude this section by the following corollary:
Corollary 3.2. Although the spaces `λp , c0, c and `∞ overlap, the space `λp does not
include any of the other spaces. Further, if lim inf
λn+1
λn
= 1 then none of the spaces
c0, c or `∞ includes the space `λp , where 0 < p <∞.
Proof. Let 0 < p < ∞. Then, it is obvious by Lemma 3.1 that the spaces `λp , c0, c
and `∞ overlap.
Further, the space `λp does not include the space c0. To show this, we define the
sequence x = (xk) ∈ c0 by
xk =
1
(k + 1)
1/p
, k ∈ N.
Then, we have for every n ∈ N that
|Λn(x)| = 1
λn
n∑
k=0
λk − λk−1
(k + 1)
1/p
≥
≥ 1
λn(n+ 1)
1/p
n∑
k=0
(λk − λk − 1) =
1
(n+ 1)
1/p
which shows that Λ(x) /∈ `p and hence x /∈ `λp . Thus, the sequence x is in c0 but not in
`λp . Hence, the space `λp does not include any of the spaces c0, c or `∞.
Moreover, if lim inf
λn+1
λn
= 1 then the space `∞ does not include the space `λp . To
see this, let 0 < p < ∞. Then, Lemma 3.4 implies that the sequence y, defined in the
proof of Theorem 3.4, is in `λp but not in `∞. Therefore, none of the spaces c0, c or `∞
includes the space `λp when lim inf
λn+1
λn
= 1, where 0 < p <∞.
Corollary 3.2 is proved.
4. The basis for the space `λp . In the present section, we give a sequence of points
of the space `λp which forms a basis for `λp , where 1 ≤ p <∞.
If a normed sequence space X contains a sequence (bn) with the property that for
every x ∈ X there is a unique sequence of scalars (αn) such that
lim
n
‖x− (α0b0 + α1b1 + . . .+ αnbn)‖ = 0,
then (bn) is called a Schauder basis (or briefly basis) for X. The series
∑
k
αkbk which
has the sum x is then called the expansion of x with respect to (bn), and written as
x =
∑
k
αkbk.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
500 M. MURSALEEN, A. K. NOMAN
Now, because of the transformation T defined from `λp to `p in the proof of Theorem
2.2 is onto, the inverse image of the basis {e(k)}k∈N of the space `p is the basis for the
new space `λp , where 1 ≤ p <∞. Therefore, we have the following:
Theorem 4.1. Let 1 ≤ p < ∞ and define the sequence e(k)(λ) = {e(k)
n (λ)}n∈N
of the elements of the space `λp for every fixed k ∈ N by
e(k)
n (λ) =
(−1)n−k
λk
λn − λn−1
, k ≤ n ≤ k + 1,
n ∈ N.
0, n < k or n > k + 1,
(4.1)
Then, the sequence {e(k)(λ)}k∈N is a basis for the space `λp and every x ∈ `λp has a
unique representation of the form
x =
∑
k
αk(λ)e(k)(λ), (4.2)
where αk(λ) = Λk(x) for all k ∈ N,
Proof. Let 1 ≤ p <∞. Then, it is clear by (4.1) that
Λ
(
e(k)(λ)
)
= e(k) ∈ `p, k ∈ N,
and hence e(k)(λ) ∈ `λp for all k ∈ N.
Further, let x ∈ `λp be given. For every non-negative integer m, we put
x(m) =
m∑
k=0
αk(λ)e(k)(λ).
Then, we have that
Λ(x(m)) =
m∑
k=0
αk(λ)Λ
(
e(k)(λ)
)
=
m∑
k=0
Λk(x)e(k)
and hence
Λn(x− x(m)) =
0, 0 ≤ n ≤ m,
n,m ∈ N.
Λn(x), n > m,
Now, for any given ε > 0, there is a non-negative integer m0 such that
∞∑
n=m0+1
|Λn(x)|p ≤
(ε
2
)p
.
Hence, we have for every m ≥ m0 that
‖x− x(m)‖`λp =
( ∞∑
n=m+1
|Λn(x)|p
)1/p
≤
( ∞∑
n=m0+1
|Λn(x)|p
)1/p
≤ ε
2
< ε.
Thus, we obtain that
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ON SOME IMBEDDING RELATIONS BETWEEN CERTAIN SEQUENCE SPACES 501
lim
m
‖x− x(m)‖`λp = 0
which shows that x ∈ `λp is represented as in (4.2).
Finally, let us show the uniqueness of the representation (4.2) of x ∈ `λp . For this,
suppose on the contrary that there exists another representation x =
∑
k
βk(λ)e(k)(λ).
Since the linear transformation T defined from `λp to `p in the proof of Theorem 2.2 is
continuous [19] (Theorem 4.2.8), we have that
Λn(x) =
∑
k
βk(λ)Λn(e(k)(λ)) =
∑
k
βk(λ)e(k)
n = βn(λ), n ∈ N,
which contradicts the fact that Λn(x) = αn(λ) for all n ∈ N. Hence, the representation
(4.2) of x ∈ `λp is unique.
Theorem 4.1 is proved.
Now, it is known by Theorem 2.1 that `λp , 1 ≤ p < ∞, is a Banach space with its
natural norm. This leads us together with Theorem 4.1 to the following corollary:
Corollary 4.1. The sequence space `λp of non-absolute type is separable for 1 ≤
≤ p <∞.
Finally, we conclude our work by expressing from now on that the aim of the next
paper is to determine the α-, β- and γ-duals of the space `λp and is to characterize some
matrix classes concerning the space `λp , where 1 ≤ p <∞.
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ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
|
| id | umjimathkievua-article-2735 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:29:17Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1e/543b03fa0236de2b56661ff1c4bc681e.pdf |
| spelling | umjimathkievua-article-27352020-03-18T19:34:55Z On some imbedding relations between certain sequence spaces Про деякi спiввiдношення вкладення мiж певними просторами послiдовностей Mursaleen, M. Noman, A. K. Мурзаліїв, М. Номан, А. К. In the present paper, we introduce the sequence space $l^{\lambda}_p$ of non-absolute type which is a $p$-normed space and a $BK$-space in the cases of $0 < p < 1$ and $0 < p < 1$ i $1 \leq p < \infty$, respectively. Further, we derive some imbedding relations and construct the basis for the space $l^{\lambda}_p$, where $1 \leq p < \infty$. Введено поняття простору послiдовностей $l^{\lambda}_p$ неабсолютного типу, який є $p$-нормованим простором i $BK$-простором у випадках $0 < p < 1$ i $1 \leq p < \infty$ вiдповiдно. Крiм того, отримано деякi спiввiдно- шення вкладення та побудовано базис для простору $l^{\lambda}_p$, де $1 \leq p < \infty$. Institute of Mathematics, NAS of Ukraine 2011-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2735 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 4 (2011); 489-501 Український математичний журнал; Том 63 № 4 (2011); 489-501 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2735/2226 https://umj.imath.kiev.ua/index.php/umj/article/view/2735/2227 Copyright (c) 2011 Mursaleen M.; Noman A. K. |
| spellingShingle | Mursaleen, M. Noman, A. K. Мурзаліїв, М. Номан, А. К. On some imbedding relations between certain sequence spaces |
| title | On some imbedding relations between certain sequence spaces |
| title_alt | Про деякi спiввiдношення вкладення мiж певними просторами послiдовностей |
| title_full | On some imbedding relations between certain sequence spaces |
| title_fullStr | On some imbedding relations between certain sequence spaces |
| title_full_unstemmed | On some imbedding relations between certain sequence spaces |
| title_short | On some imbedding relations between certain sequence spaces |
| title_sort | on some imbedding relations between certain sequence spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2735 |
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