On the sequence space 𝑙_{(𝑝_{𝑛})} and 𝜆_{(𝑝_{𝑛})}, 0<𝑝_{𝑛}≤1
Authors: Schonefeld S. A. , Stiles W. J. 
Published: July 2010

Let

(

p
n

)

({p_n})

and

(

q
n

)

({q_n})

be sequences in the interval

(
0
,
1
]

(0,1]

, let

l

(

p
n

)

{l_{({p_n})}}

be the set of all real sequences

(

x
n

)

({x_n})

such that

∑

|

x
n

|

p
n

>
∞

\sum {|{x_n}{|^{{p_n}}} > \infty }

, and let

λ

(

q
n

)

{\lambda _{({q_n})}}

be the set of all real sequences

(

y
n

)

({y_n})

such that

sup
π

∑

|

y
n

|

q

π
(
n
)

>
∞

{\sup _\pi }\sum {|{y_n}{|^{{q_{\pi (n)}}}} > \infty }

where the sup is taken over all permutations

π
\pi

of the positive integers. The purpose of this paper is to investigate some of the properties of these spaces. Our results are primarily concerned with (1) conditions which are necessary and/or sufficient for

l

(

p
n

)

{l_{({p_n})}}

(resp.,

λ

(

p
n

)

{\lambda _{({p_n})}}

) to equal

l

(

q
n

)

{l_{({q_n})}}

(resp.,

λ

(

q
n

)

{\lambda _{({q_n})}}

), and (2) isomorphic and topological properties of the subspaces of these spaces. In connection with (1), we show that the following four conditions are equivalent for any sequence

(

ε
n

)

({\varepsilon _n})

which decreases to zero and has

ε
1

>
1

{\varepsilon _1} > 1

. (a) There exists a number

K
>
1

K > 1

such that the series

∑

1

/

K

1

/

ε
n

\sum {1/{K^{1/{\varepsilon _n}}}}

converges; (b) the elements

ε
n

{\varepsilon _n}

of the sequence satisfy the condition

ε
n

=
O
(
1

/

ln
⁡
n
)

{\varepsilon _n} = O(1/\ln n)

; (c) the sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-parenthesis ln n right-parenthesis left-parenthesis left-parenthesis 1 slash n right-parenthesis sigma-summation Underscript 1 Overscript n Endscripts epsilon Subscript j Baseline right-parenthesis right-parenthesis">

(
(
ln
⁡
n
)
(
(
1

/

n
)

∑
1
n

ε
j

)
)

((\ln n)((1/n)\sum \nolimits _1^n {{\varepsilon _j}} ))

is bounded; and (d)

l

(
1
−

ε
n

)

{l_{(1 - {\varepsilon _n})}}

equals

l
1

{l_1}

. In connection with (2), we show that the following are true when

(

p
n

)

({p_n})

increases to one. (a)

λ

(

p
n

)

{\lambda _{({p_n})}}

contains an infinite-dimensional closed subspace where the

l

(

p
n

)

{l_{({p_n})}}

-topology and the

λ

(

p
n

)

{\lambda _{({p_n})}}

-topology agree; (b)

l

(

p
n

)

{l_{({p_n})}}

and

λ

(

p
n

)

{\lambda _{({p_n})}}

contain closed subspaces isomorphic to

l
1

{l_1}

; and (c)

λ

(

p
n

)

{\lambda _{({p_n})}}

contains no infinite-dimensional subspace where the

λ

(

p
n

)

{\lambda _{({p_n})}}

-topology agrees with the

l
1

{l_1}

-topology if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit left-parenthesis left-parenthesis 1 slash n right-parenthesis Superscript p 1 Baseline plus left-parenthesis 1 slash n right-parenthesis Superscript p 2 Baseline plus midline-horizontal-ellipsis plus left-parenthesis 1 slash n right-parenthesis Superscript p Super Subscript n Superscript Baseline right-parenthesis equals normal infinity period">

lim
(

(
1

/

n

)

p
1

+

(
1

/

n

)

p
2

+
⋯
+

(
1

/

n

)

p
n

)
=
∞
.

\lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .

\]