On some imbedding relations between certain sequence spaces
In the present paper, we introduce the sequence space ` lλp 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 lλp, where 1 ≤ p < ∞....
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irk-123456789-1660252020-02-19T01:26:06Z On some imbedding relations between certain sequence spaces Mursaleen, M. Noman, A.K. Статті In the present paper, we introduce the sequence space ` lλp 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 lλp, where 1 ≤ p < ∞. Remove selected Введено поняття простору послiдовностей lλp неабсолютного типу, який є p-нормованим простором i BK-простором у випадках 0 < p < 1 i 1 ≤ p < ∞ вiдповiдно. Крiм того, отримано деякi спiввiдношення вкладення та побудовано базис для простору lλp, де 1 ≤ p < ∞. 2011 Article On some imbedding relations between certain sequence spaces / M. Mursaleen, A.K. Noman // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 489–501. — Бібліогр.: 19 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166025 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Mursaleen, M. Noman, A.K. On some imbedding relations between certain sequence spaces Український математичний журнал |
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In the present paper, we introduce the sequence space `
lλp 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 lλp, where 1 ≤ p < ∞.
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Mursaleen, M. Noman, A.K. |
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Mursaleen, M. Noman, A.K. |
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Mursaleen, M. |
| title |
On some imbedding relations between certain sequence spaces |
| title_short |
On some imbedding relations between certain sequence spaces |
| title_full |
On some imbedding relations between certain sequence spaces |
| title_fullStr |
On some imbedding relations between certain sequence spaces |
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On some imbedding relations between certain sequence spaces |
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on some imbedding relations between certain sequence spaces |
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Інститут математики НАН України |
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2011 |
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Статті |
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| citation_txt |
On some imbedding relations between certain sequence spaces / M. Mursaleen, A.K. Noman // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 489–501. — Бібліогр.: 19 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT mursaleenm onsomeimbeddingrelationsbetweencertainsequencespaces AT nomanak onsomeimbeddingrelationsbetweencertainsequencespaces |
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2025-07-14T20:30:27Z |
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2025-07-14T20:30:27Z |
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1837655683887005696 |
| 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
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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
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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.
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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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Received 10.08.09,
after revision — 18.06.10
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