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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 &gt; ∞  \sum {|{x_n}{|^{{p_n}}} &gt; \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 ) &gt; ∞  {\sup _\pi }\sum {|{y_n}{|^{{q_{\pi (n)}}}} &gt; \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 &gt; 1 {\varepsilon _1} &gt; 1      . (a) There exists a number      K &gt; 1 K &gt; 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      ( ( 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  \[    lim ( ( 1 / n ) p 1 + ( 1 / n ) p 2 + ⋯  + ( 1 / n ) p n ) = ∞  . \lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .     \]

On the sequence space 𝑙_{(𝑝_{𝑛})} and 𝜆_{(𝑝_{𝑛})}, 0<𝑝_{𝑛}≤1