16 B. de Malafosse
2.1. Properties of the sequence C α α.
We shall use the following sets c
C
1
= α ∈
U
+∗
1 α
n n
X
k=1
α
k
= O 1 n → ∞
, b
C =
α ∈
U
+∗
1 α
n n
X
k=1
α
k n≥1
∈ c
,
c C
+ 1
= α ∈
U
+∗
\ cs
1 α
n ∞
X
k=n
α
k
= O 1 n → ∞
, Ŵ
= α ∈
U
+∗
lim
n→∞
α
n−1
α
n
1 and
Ŵ
+
= α ∈
U
+∗
lim
n→∞
α
n+1
α
n
1 .
Note that α ∈ Ŵ
+
if and only if
1 α
∈ Ŵ . We shall see in Proposition 1 that if α ∈ c
C
1
, α tends to infinity. On the other hand we see that 1 ∈ Ŵ
α
implies α ∈ Ŵ. We also have α ∈ Ŵ
if and only if there is an integer q ≥ 1 such that γ
q
α = sup
n≥q+1
α
n−1
α
n
1. We obtain the following results in which we put [C α α]
n
= 1
α
n n
P
k=1
α
k
. P
ROPOSITION
1. Let α ∈ U
+∗
. Then i
α
n−1
α
n
→ 0 if and only if [C α α]
n
→ 1.
ii [C α α]
n
→ l implies that
α
n−1
α
n
→ 1 −
1 l
. iii If α ∈ c
C
1
then there are K 0 and γ 1 such that α
n
≥ K γ
n
for all n. iv α ∈ Ŵ implies that α ∈ c
C
1
and there exist a real b 0 and an integer q, such that
[C α α]
n
≤ 1
1 − χ +
bχ
n
for n ≥ q + 1 and χ = γ
q
α ∈]0, 1[. v α ∈ Ŵ
+
implies α ∈ c C
+ 1
.
On the sets of sequences 17
Proof. i, ii, iii and iv have been proved in [10]. Assertion v. If α ∈ Ŵ
+
, there are χ
′
∈ ]0, 1[ and an integer q
′
≥ 1 such that
α
k
α
k−1
≤ χ
′
for k ≥ q
′
. Then we have for every n ≥ q
′
1 α
n ∞
X
k=n
α
k
=
∞
X
k=n
α
k
α
n
≤ 1 +
∞
X
k=n+1 k−n−1
Y
i=0
α
k−i
α
k−i−1
≤
∞
X
k=n
χ
′ k−n
= O 1 .
This gives the conclusion. R
EMARK
1. Note that as a direct consequence of Proposition 2.1, we have b
C ⊂ Ŵ ⊂ c C
1
. We also have b
C 6= Ŵ, see [4]. On the other hand we see that c C
1
T c C
+ 1
= Ŵ T
Ŵ
+
= φ .
2.2. The spaces w
p α
λ, w
◦
p α
λ and w
• p
α
λ for p 0.
In this subsection we recall some results on the sets that generalize the sets w
p ∞
λ, w
p
λ and w
p
λ for given real p 0. For any given real p 0 and every sequence X = x
n n≥1
, we put |X |
p
= x
p n
n
and w
p α
λ =
X ∈ s C λ |X |
p
∈ s
α
, w
◦
p α
λ =
n X ∈ s C λ |X |
p
∈ s
◦
α
o ,
w
• p
α
λ =
n X ∈ s X − le
t
∈ w
◦
p α
λ for some l ∈ C
o .
For instance we see that w
p α
λ = X = x
n n
∈ s sup
n≥1
1 |λ
n
| α
n n
X
k=1
|x
k
|
p
∞ .
If there exist A and B 0, such that A α
n
B for all n, we get the well known spaces w
p α
λ = w
p ∞
λ, w
◦
p α
λ = w
p
λ and w
• p
α
λ = w
p
λ, see [14, 15]. In the case when λ = n
n≥1
, the previous sets have been introduced in [3] by Maddox and it is written w
p ∞
λ = w
p ∞
, w
p
λ = w
p
and w
p
λ = w
p
. It is proved that each of the sets w
p
and w
p ∞
is a p−normed FK space for 0 p 1, that is a complete linear metric space in which each projection P
n
is continuous, and a BK space for
18 B. de Malafosse
1 ≤ p ∞ with respect to the norm
kX k =
sup
ν≥ 1
1 2
ν
2
ν+ 1
− 1
P
n=2
ν
|x
n
|
p
if 0 p 1, sup
ν≥ 1
1 2
ν
2
ν+ 1
− 1
P
n=2
ν
|x
n
|
p
1 p
if 1 ≤ p ∞. w
p
has the property AK, and every sequence X = x
n n≥1
∈ w
p
n
n
has a unique representation
X = le
t
+
∞
X
n=1
x
n
− l e
t n
,where l ∈ C is such that X − le
t
∈ w
p
, When p = 1, we omit the index p and write w
p α
λ = w
α
λ, w
◦
p α
λ = w
◦
α
λ and w
• p
α
λ = w
α
λ. It has been proved in [14], that if λ is a strictly increasing sequence of reals tending to infinity then w
λ and w
∞
λ are BK spaces and w λ has AK,
with respect to the norm kX k = kC λ |X |k
l
∞
= sup
n
1 λ
n n
X
k=1
|x
k
| .
Recall the next results given in [10]. T
HEOREM
1. Let α and λ be any sequences of U
+∗
. i Consider the following properties
a
α
n−1
λ
n−1
α
n
λ
n
→ 0;
b s
• α
C λ = s
• αλ
. c αλ ∈ c
C
1
; d w
α
λ = s
αλ
; e w
◦
α
λ = s
◦
αλ
; f w
• α
λ = s
◦
αλ
. We have a⇒b, c⇔d and c⇒e and f.
ii If αλ ∈ c C
1
, w
α
λ, w
◦
α
λ and w
• α
λ are BK spaces with respect to the norm kX k
s
αλ
= sup
n≥1
|x
n
| α
n
λ
n
, and w
◦
α
λ = w
• α
λ has AK.
On the sets of sequences 19
2.3. Properties of some new sets of sequences.
Kategori