Internat. J. Math. & Math. Sci.
Vol. 24, No. 2 (2000) 121–128
S0161171200002453
© Hindawi Publishing Corp.
THE SECOND DUAL SPACES OF THE SETS OF Λ-STRONGLY
CONVERGENT AND BOUNDED SEQUENCES
A. M. JARRAH and E. MALKOWSKY
(Received 19 November 1998)
p
p
Abstract. We give the second β-, γ-, and f -duals of the sets w0 (Λ), w∞ (Λ) (0 < p < ∞),
p
p
p
c0 (Λ), c p (Λ), and c∞ (Λ) (0 < p ≤ 1) and the second continuous dual spaces of w0 (Λ),
p
p
c0 (Λ), and c p (Λ) for 0 < p ≤ 1. Furthermore, we determine the α-duals of c0 (Λ), c p (Λ),
p
and c∞ (Λ) for 1 < p < ∞.
Keywords and phrases. BK spaces, AK spaces, AD spaces, dual spaces.
2000 Mathematics Subject Classification. Primary 40H05, 46A45; Secondary 47B07.
1. Introduction and well-known results. We write ω for the set of all complex
sequences x = (xk )∞
k=0 , φ, l∞ , c and c0 for the sets of all finite, bounded, convergent
sequences, and sequences convergent to naught, respectively, further cs, bs, and l1
for the sets of all convergent, bounded, and absolutely convergent series.
By e and e(n) (n ∈ N0 ), we denote the sequences such that ek = 1 for k = 0, 1, . . . , and
n
(n)
(n)
[n]
en = 1 and ek = 0 for k = n. For any sequence x = (xk )∞
= k=0 xk e(k)
k=0 , let x
be its n-section.
Let X, Y ⊂ ω and z ∈ ω. Then we write
z−1 × X = x ∈ ω : xz = (xk zk )∞
k=0 ∈ X ,
M(X, Y ) =
x −1 × Y = a ∈ ω : ax ∈ Y ∀x ∈ X
(1.1)
x∈X
for the multiplier space of X and Y . The sets M(X, l1 ), M(X, cs), and M(X, bs) are
called the α-, β-, and γ-duals of X.
A Fréchet subspace X of ω is called an FK space if it has continuous coordinates,
that is, if convergence in X implies coordinatewise convergence. An FK space X ⊃ φ
[n]
→ x (n → ∞); and it is
is said to have AK if, for every sequence x = (xk )∞
k=0 ∈ X, x
said to have AD if φ is dense in X. A BK space is an FK space which is a Banach space.
If X is a p-normed space, then we write X ∗ for the set of all continuous linear
functionals on X, the so-called continuous dual of X, with its norm · is given by
f = sup |f (x)| : x = 1
∀f ∈ X ∗ .
(1.2)
∗
Let X ⊃ φ be an FK space. Then the set X f = {(f (e(n) ))∞
n=0 : f ∈ X } is called the
f -dual of X.
Given any infinite matrix A = (ank )∞
n,k=0 of complex numbers and any sequence
∞
x ∈ ω, let An (x) = k=0 ank xk (n = 0, 1, . . . ), and let A(x) = (An (x))∞
n=0 provided the
122
A. M. JARRAH AND E. MALKOWSKY
series converge, and XA = x ∈ ω : A(x) ∈ X . If 0 < p < ∞, then we write |x|p =
p
p
(|xk |p )∞
k=0 and X[A] = {x ∈ ω : A(|x| ) ∈ X}.
∞
Let 0 < p < ∞ and µ = (µn )n=0 be a nondecreasing sequence of positive integers
tending to infinity, throughout. We define the matrices ∆ and M by


1
(k = n),



∆nk = −1 (k = n − 1),




0
(otherwise),
(1.3)

1

 p (0 ≤ k ≤ n)
(n = 0, 1, . . . )
Mnk = µn


0
(k > n)
and use the convention that any symbol with a negative subscript has the value zero.
The sets
p
p
w0 (µ) = c0 [M]p ,
w∞
(µ) = l∞ [M]p ,
p
p
c0 (µ) = (µ)−1 × w0 (µ) ∆ ,
p
p
c∞
(µ) = (µ)−1 × w∞
(µ) ∆ ,
p
c p (µ) = c0 (µ) ⊕ e
(1.4)
were studied in [1], and their first duals were given there. If p = 1, then we omit the
index p, i.e., we write w0 (µ) = w01 (µ), etc.
Following the notation introduced in [3], we say that a nondecreasing sequence Λ =
(λn )∞
n=0 of positive reals tending to infinity is exponentially bounded if there are reals
s and t with 0 < s ≤ t < 1 such that for some subsequence (λn(ν) )∞
ν=0 of Λ, we have
s≤
λn(ν)
≤t
λn(ν+1)
∀ν = 0, 1, . . . ;
(1.5)
such a subsequence (λn(ν) )∞
ν=0 is called an associated subsequence.
If (n(ν))∞
ν=0 is a strictly increasing sequence of nonnegative integers, then we write
K ν for the set of all integers k with n(ν) ≤ k ≤ n(ν + 1) − 1, and ν and maxν for
the sum and maximum taken over all k in K ν .
p
p
p
p
If X p (Λ) denotes any of the sets w0 (Λ), w∞ (Λ), c0 (Λ), c p (Λ), or c∞ (Λ), then we
p
n
write X̃ p (Λ) for the respective space with the sections 1/λn k=0 . . . replaced by the
p
blocks 1/λn(ν+1) ν . . . . Furthermore, we define

n


p
sup 1p
|x
|
(0 < p ≤ 1),

k

 n
λn k=0
p
1/p
xw∞
n
(Λ) = 

1 
p

|x
|
(1 ≤ p < ∞),
sup

k
p
 n
λ
n k=0

1

p


|xk |

p
sup
λn(ν+1) ν
ν
1/p
p
=
xw̃∞
(Λ)


1

p

|xk |
sup
p
λn(ν+1) ν
ν
p
p
xc∞
(Λ) = ∆(Λx)w∞ (Λ) ,
(0 < p ≤ 1),
(1 ≤ p < ∞),
p
p
xc̃∞
(Λ) = ∆(Λx)w̃∞ (Λ) .
(1.6)
123
THE SECOND DUAL SPACES OF THE SETS OF Λ-STRONGLY . . .
p
p
2. The second duals of the sets w0 (Λ) and w∞ (Λ) for 0 < p < ∞. Let Λ = (λn )∞
n=0
be a nondecreasing exponentially bounded sequence of positive reals throughout and
let (λn(ν) )∞
ν=0 be an associated subsequence. We put

∞


ak < ∞
 a∈ω:
λ
max
(0 < p ≤ 1),

n(ν+1)

ν

ν=0
p
ᐃ (Λ) = 
∞
p 1/p

p



a
∈
ω
:
a
λ
<
∞
1
<
p
<
∞,
q
=
n(ν+1)
k

p −1
ν
ν=0
(2.1)
and, on ᐃp (Λ),
 ∞


 λn(ν+1) max ak 

ν
ν=0
aᐃp (Λ) = ∞

p 1/p


ak 
 λn(ν+1)

(0 < p ≤ 1),
p
.
1 < p < ∞, q =
p −1
ν
ν=0
p
(2.2)
p
In [1, Theorem 2], it was shown that if X p (Λ) = w0 (Λ) or X p (Λ) = w∞ (Λ) and †
p
stands for α, β, γ, or f , then (X p (Λ))† = ᐃp (Λ), that the continuous dual (w0 (Λ))∗
p
p
p
of w0 (Λ) is norm isomorphic to ᐃp (Λ) when w0 (Λ) has the norm · w̃∞
(Λ) , and
p
= aᐃp (Λ) on (w∞ (Λ))β . Furthermore, ᐃp (Λ) is a BK space with
finally that a∗
p
w̃ (Λ)
∞
AK with ·ᐃp (Λ) (cf. [2]). Therefore the following result gives the second duals of the
p
p
sets w0 (Λ) and w∞ (Λ).
Theorem 2.1. We put p = max{1, p}. If † stands for any of the symbols α, β, γ,
p
or f , then (ᐃp (Λ))† = w∞ (Λ) for 0 < p < ∞, and the continuous dual (ᐃp (Λ))∗ of
p
ᐃp (Λ) is norm isomorphic to w∞ (Λ) with · p
w̃∞
.
Proof. The statements of the theorem with the exception of those concerning the
γ- and f -duals are well known (cf. [2, Theorems 2, 4, 5, and 6]).
Since ᐃp (Λ) has AK, it follows that (ᐃp (Λ))β = (ᐃp (Λ)) f by [4, Theorem 7.2.7(ii),
p
page 106], and so (ᐃp (Λ)) f = w∞ (Λ). Further ᐃp (Λ) has AD, since it has AK, and so
p
(ᐃp (Λ))β = (ᐃp (Λ))γ by [4, Theorem 7.2.7(iii), page 106], hence (ᐃp (Λ))γ = w∞ (Λ).
p
3. The α-duals of the sets c0 (Λ), c p (Λ), and c p (Λ) for 1 < p < ∞
Theorem 3.1. We put
p
Ꮿα (Λ)
= a∈ω:
∞
λn(ν+1)
ν=0
a
p
Ꮿα (Λ)
=
ν
∞ ak q 1/q
k=n
λk
∞
λn(ν+1)
ν=0
p
ν
<∞
1 < p < ∞; q =
∞ ak q 1/q
.
λk
k=n
p
p
,
p −1
(3.1)
p
If X p (Λ) denotes any of the sets c0 (Λ), c p (Λ), and c∞ (Λ), then (X p (Λ))α = Ꮿα (Λ).
p
Furthermore, Ꮿα (Λ) is a BK space with · Ꮿpα (Λ) .
124
A. M. JARRAH AND E. MALKOWSKY
p
p
Proof. First, we assume a ∈ Ꮿα (Λ), and let x ∈ c∞ (Λ). Then there is a constant
M such that
1/p
∆(Λx) p
≤ λn(ν+1) M ∀ν = 0, 1, . . . .
(3.2)
n
Putting Rn =
∞
ν
k=n (|ak |/λk )
(n = 0, 1, . . . ) and using Hölder’s inequality, we obtain
k
∞
∞ ∞
∞ ak ak ak xk ≤
∆(Λx) =
∆(Λx) n
n
λk n=0
λk
ν=0
n=0
k=0
k=n
1/q
∞
∞
q
∆(Λx) Rn ≤ M
=
λn(ν+1)
Rn
.
n
ν=0 ν
This shows that
p
Ꮿα (Λ)
⊂
p
(c∞ (Λ))α
∞
(3.3)
ν
ν=0
and that
ak xk ≤ a
k=0
Ꮿα (Λ) xc̃∞ (Λ) .
p
(3.4)
p
p
(m)
p
(m)
Conversely, we assume a ∈ c0 (Λ). We define the maps fa : c0 (Λ) → R by fa (x) =
m
p
(m) ∞
)m=0 is a sequence of seminorms on c0 (Λ) which
k=0 |ak xk | (x ∈ X). Then (fa
p
(m)
are continuous, since c0 (Λ) is a BK space by [1, Theorem 1]. Further, fa (x) ≤
∞
k=0 |ak xk | = M(x) < ∞ for all m ∈ N0 and for all x ∈ X. By the uniform boundedness principle, there is a constant M such that
∞
|ak xk | ≤ M
k=0
p
p
∀x ∈ c0 (Λ) with xc̃∞
(Λ) ≤ 1.
p
(3.5)
p
Since a ∈ (c0 (Λ))α and 1/Λ = (1/λk )∞
k=0 ∈ c0 (Λ), the numbers Rn are defined for all
n. We put
n(µ+1)−1
Sµ =
q
Rl
l=n(µ)
∀µ = 0, 1, . . . .
Let ν(m) ∈ N0 be given. We define the sequence x ν(m) by



n(µ+1)−1
ν−1
n


1 
−1/p
q−1
−1/p
q−1 


λn(µ+1) Sµ
Rk + λn(ν+1) Sν
Rk



λn µ=0

k=n(µ)
k=n(ν)


ν(m)
=
xn
n ∈ N ν ; 0 ≤ ν ≤ ν(m) ,



n(µ+1)−1
ν(m)


1 
−1/p
q−1 

λn(µ+1) Sµ
Rk
n ≥ n ν(m) + 1 .


λn µ=0
k=n(µ)
Then

q−1

λn(ν+1) Sν−1/p Rn
ν(m)
∆ Λx
n=
0
∆ Λx ν(m)
n =
ν
(3.7)
n ∈ N ν ; ν = 0, 1, . . . , ν(m) ,
n ∈ N ν ; ν ≥ ν(m) + 1 ,
 p
q
p

λn(ν+1) Sν−1 Rn = λn(ν+1)

0
ν
(3.6)
0 ≤ ν ≤ ν(m) ,
ν ≥ ν(m + 1) .
(3.8)
125
THE SECOND DUAL SPACES OF THE SETS OF Λ-STRONGLY . . .
p
ν(m)
p
Thus x ν(m) ∈ c0 (Λ) and x ν(m) c̃∞
(Λ) = 1. Now, by (3.5) and (3.8) and since xk
for all k = 0, 1, . . . ,
ν(m)
λn(ν+1)
ν
ν=0
q
Rn
1/q
=
ν(m)
λn(ν+1)
ν=0
q
Rn
−1/p
Sν
ν
=
ν(m)
ν=0 ν
−1/p
λn(ν+1) Sν
q−1
Rn
≥0
Rn
∞
∞ ak ∆ Λx ν(m)
Rn ≤
∆ Λx ν(m)
n
n
λk
ν=0 ν
n=0
k=n
k
∞ ∞ ak ak ν(m) =
λk xk
∆ Λx ν(m) n =
λ
λ
k
k
n=0
k=0
k=0
=
=
ν(m)
∞
ν(m) ak x
≤ M.
k
k=0
(3.9)
Since ν(m) ∈ N0 was arbitrary, we have
∞
λn(ν+1)
ν=0
q
Rn
ν
1/q
≤
∞
ak xk < ∞,
(3.10)
k=0
p
that is, a ∈ Ꮿα (Λ).
p
p
p
p
Therefore we have shown (c∞ (Λ))α = (c0 (Λ))α = Ꮿα (Λ). Since c0 (Λ) ⊂ c p (Λ) ⊂
p
p
p
α
c∞ (Λ) for 1 < p < ∞ (cf. [1, Lemma 1(b)]), we also have (c (Λ)) = Ꮿα (Λ).
p
Finally, Ꮿα (Λ) is a BK space with ·Ꮿpα (Λ) by [4, Theorem 4.3.15, page 64], (3.4), and
(3.10).
p
p
4. The second duals of the sets c0 (Λ), c p (Λ), and c∞ (Λ) for 0 < p ≤ 1. We put
∞
∞
a k
<∞ ,
Ꮿ(Λ) = a ∈ ω :
λn(ν+1) max ν λ ν=0
k=n k
(4.1)
∞
∞
ak .
λn(ν+1) max aᏯ(Λ) =
ν λk ν=0
k=n
p
p
In [1, Theorem 4], it was shown that if X p (Λ) is any of the sets c0 (Λ) or c∞ (Λ) and
† stands for any of the symbols β, γ, or f , then (X p (Λ))† = Ꮿ(Λ) and that this also
holds when X p (Λ) = c(Λ) or X p (Λ) = c p (Λ) for 0 < p < 1 whenever
sup
n
n
1 ∆(µx) p < ∞;
p
k
µn k=0
(4.2)
otherwise (c p (Λ))β = Ꮿ(Λ)∩cs and (c p (Λ))γ = Ꮿ(Λ)∩bs. Furthermore, it was shown
p
p
p
that the continuous dual (c0 (Λ))∗ of c0 (Λ) is norm isomorphic to Ꮿ(Λ) when c0 (Λ)
p
∗
p
has the p-norm · c̃∞
= aᏯ(Λ) on c∞ (Λ). Finally, that f ∈ c ∗ (Λ)
p
(Λ) and ac̃∞
(Λ)
∞
if and only if f (x) = lχf + n=0 an xn for all x ∈ c(Λ) where a ∈ Ꮿ(Λ), l ∈ C with
∞
x − le ∈ c0 (Λ) and χf = f (e) − n=0 an , and that f is equivalent to |χf | + aᏯ(Λ) ;
if condition (4.2) is satisfied, then this also holds for c p (Λ) (0 < p < 1).
Therefore the following result gives the second duals.
126
A. M. JARRAH AND E. MALKOWSKY
Theorem 4.1. (a) The space Ꮿ(Λ) with · Ꮿ(Λ) is a BK space with AK.
ββ
(b) The set c∞ (Λ) is β perfect, that is, c∞ (Λ) = c∞ (Λ). Further a∗
Ꮿ(Λ) = ac̃∞ (Λ) for
β
all a ∈ Ꮿ (Λ).
(c) Finally, Ꮿf (Λ) = Ꮿγ (Λ) = Ꮿβ (Λ).
Proof. We apply Abel’s summation by parts. If a ∈ cs, then we write R(a) for the
∞
sequence with Rn (a) = k=n ak (n = 0, 1, . . . ). Then
m−1
an yn =
n=0
m
Rn (a)(∆y)n − Rm (a)ym
∀m = 0, 1, . . . .
(4.3)
n=0
(a) The space ᐃ(Λ) is a BK space with · ᐃ(Λ) (cf. [2, Theorem 2]). Further, the
matrix A defined by ank = 1/λk for k ≥ n and ank = 0 for 0 ≤ n − 1 (n = 0, 1, . . . ) is
one-to-one, and x = A(y) ∈ Ꮿ(Λ) if and only if y ∈ ᐃ(Λ). So, by [4, Theorem 4.3.2,
page 61], Ꮿ(Λ) is a BK space with xᏯ(Λ) = A(y)ᐃ(Λ) . Now, we show that Ꮿ(Λ) has
AK. First, we observe that φ ⊂ Ꮿ(Λ), since Ꮿ(Λ) is the β-dual of a sequence space. Now,
let x ∈ Ꮿ(Λ) and let ε > 0. For each m ∈ N0 , let νm denote the uniquely determined
integer for which m ∈ N νm . We choose m0 ∈ N0 such that
∞
ν=νm
λn(ν+1) max Rn x/Λ < ε
ν
∀m ≥ m0 .
(4.4)
Let m ≥ m0 . Since the sequence Λ = (λn )∞
n=0 is exponentially bounded, there is t ∈
(0, 1) such that, by (1.5),
x − x [m] Ꮿ(Λ)
=
∞
λn(ν+1) max Rn x − x [m] /Λ ν
ν=0
≤
νm
−1
∞
λn(ν+1) Rm+1 x/Λ +
λn(ν+1) max Rn x/Λ ν=νm
ν=0
< ε+
νm
−1
ν=0
< ε+ε
ν
λn(ν+1)
λn(νm +1) maxRn x/Λ νm
λn(νm +1)
νm
−1
ν=0
t νm −v < ε
(4.5)
1
.
1−t
This shows that Ꮿ(Λ) has AK.
(b) First, we show that Ꮿβ (Λ) = c∞ (Λ).
For any X ⊂ ω, X ⊂ X ββ by [4, Theorem 7.1.2, page 105]. So we have to show c∞ (Λ) ⊂
β
Ꮿ (Λ) by [1, Theorem 4].
∞
Let a ∈ Ꮿβ (Λ). We define fa : Ꮿ(Λ) → C by fa (x) = k=0 ak xk for all x ∈ Ꮿ(Λ). Then
fa ∈ Ꮿ∗ (Λ) by [4, Theorem 7.2.9, page 107], and so
fa (x) ≤ fa xᏯ(Λ) < ∞
∀x ∈ Ꮿ(Λ).
(4.6)
THE SECOND DUAL SPACES OF THE SETS OF Λ-STRONGLY . . .
127
Let m ∈ N0 be given and νm the uniquely determined integer such that m ∈ N νm .
Since Λ is exponentially bounded, there are s, t ∈ (0, 1) such that, by (1.5),
(m) e Ꮿ(Λ) =
ν
m
λn(ν+1)
e(m) =
λn(ν+1) max Rn
ν
Λ
λm
k
ν=0
ν=0
∞
ν
m
νm
λn(ν+1) λn(νm +1)
1 νm −ν
1
< ∞.
≤
≤
t
≤
λ
λ
s
s(1
− t)
n(νm )
ν=0 n(νm +1)
ν=0
(4.7)
Now (4.6) implies
am = fa e(m) ≤ fa e(m) Ꮿ(Λ) ≤ fa
1
s(1 − t)
∀m ∈ N0 ,
(4.8)
and so a ∈ l∞ . Further, x ∈ Ꮿ(Λ) implies that Rn (x/Λ) ∈ cs for all n, and ΛR(x/Λ) ∈
c0 . Therefore aΛR(x/Λ) ∈ c0 . Now (4.3) yields
∞
an xn =
n=0
∞
n=0
Rn (x/Λ) ∆(Λa) n
∀x ∈ Ꮿ(Λ).
(4.9)
Thus R(x/Λ)∆(Λa) ∈ cs for all x ∈ Ꮿ(Λ). Now x ∈ Ꮿ(Λ) if and only if R(x/Λ) ∈ ᐃ(Λ)
and, by [2, Theorem 4], ∆(Λa) ∈ ᐃβ (Λ) = w∞ (Λ). But this means that a ∈ c∞ (Λ). Thus
we have shown that Ꮿβ (Λ) ⊂ c∞ (Λ).
Now we show
a∗
Ꮿ(Λ) = ac̃∞ (Λ)
∀a ∈ Ꮿβ (Λ).
(4.10)
Let a ∈ Ꮿβ (Λ) = c∞ (Λ), by what we have just shown. Then by (4.9), for all x ∈ Ꮿ(Λ),
∞
∞
1
∆(Λa) an xn ≤
λn(ν+1) max Rn (x/Λ)
n
ν
λn(ν+1)
n=0
ν
ν=0
(4.11)
≤ ac̃∞ (Λ) xᏯ(Λ) ,
and so
a∗
Ꮿ(Λ) ≤ ac̃∞ (Λ) .
(4.12)
Let νm ∈ N0 . By ν0,m , we denote the smallest integer with 0 ≤ ν0,m ≤ νm for which
1
∆(Λa) = max
n
λn(ν0,m +1) ν0,m
0≤ν≤νm
1
λn(ν+1)
∆(Λa) .
n
(4.13)
ν
We define the sequences R (m) and x (m) by
(m)
Rn
=




1
sgn ∆(Λa) n
λn(ν0,m +1)


0
for n ∈ N ν0,m ,
for n ∉ N ν0,m ,
(4.14)
128
A. M. JARRAH AND E. MALKOWSKY
(m)
and xn
(m)
= Rn
(m) x
(m)
− Rn+1 (n = 0, 1, . . . ). Then we have
Ꮿ(Λ)
=
∞
ν=0
(m) (m) = λn(ν0,m +1) max Rn
≤ 1,
λn(ν+1) max Rn
ν
ν0,m
∞
1
(m) ∆(Λa) ≤ a∗ xᏯ(Λ) ≤ a∗ .
= max
an xn
n
Ꮿ(Λ)
Ꮿ(Λ)
0≤ν≤νm λn(ν+1)
ν
n=0
(4.15)
Since νm was arbitrary, we obtain ac̃∞ (Λ) ≤ a∗
Ꮿ(Λ) . Together with (4.12), this yields
(4.10).
(c) Since Ꮿ(Λ) has AK by part (b) and so AD, part (c) follows from [4, Theorem 7.2.7(ii)
and (iii), page 106].
References
[1]
[2]
[3]
[4]
A. M. Jarrah and E. Malkowsky, The dual spaces of Λ-strongly convergent and bounded
sequences, to appear.
E. Malkowsky, Matrix Transformations in a New Class of Sequences that Includes Spaces
of Absolutely and Strongly Summable Sequences, Habilitationsschrift, Giessen, Univ.
Giessen, 1989. Zbl 684.46011.
, The continuous duals of the spaces c0 (Λ) and c(Λ) for exponentially bounded sequences Λ, Acta Sci. Math. (Szeged) 61 (1995), no. 1-4, 241–250. MR 97d:46005.
Zbl 846.40008.
A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies, 85, North-Holland Publishing Co., Amsterdam-New York, 1984, Notas de Matematica [Mathematical Notes], 91. MR 85d:40006. Zbl 531.40008.
A. M. Jarrah: Department of Mathematics, Faculty of Science, Yarmouk University,
Irbid, Jordan
E-mail address: ajarrah@yu.edu.jo
E. Malkowsky: c/o Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan
E-mail address: malkowsky@math.uni-giessen.de