On the sequence space $l_{(p_{n})}$ and $\lambda _{(p_{n})},$ $0<p_{n}\leq 1$

Abstract:

Let $({p_n})$ and $({q_n})$ be sequences in the interval $(0,1]$, let ${l_{({p_n})}}$ be the set of all real sequences $({x_n})$ such that $\sum {|{x_n}{|^{{p_n}}} < \infty }$, and let ${\lambda _{({q_n})}}$ be the set of all real sequences $({y_n})$ such that ${\sup _\pi }\sum {|{y_n}{|^{{q_{\pi (n)}}}} < \infty }$ where the sup is taken over all permutations $\pi$ 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 ${l_{({p_n})}}$ (resp., ${\lambda _{({p_n})}}$) to equal ${l_{({q_n})}}$ (resp., ${\lambda _{({q_n})}}$), 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 $({\varepsilon _n})$ which decreases to zero and has ${\varepsilon _1} < 1$. (a) There exists a number $K > 1$ such that the series $\sum {1/{K^{1/{\varepsilon _n}}}}$ converges; (b) the elements ${\varepsilon _n}$ of the sequence satisfy the condition ${\varepsilon _n} = O(1/\ln n)$; (c) the sequence $((\ln n)((1/n)\sum \nolimits _1^n {{\varepsilon _j}} ))$ is bounded; and (d) ${l_{(1 - {\varepsilon _n})}}$ equals ${l_1}$. In connection with (2), we show that the following are true when $({p_n})$ increases to one. (a) ${\lambda _{({p_n})}}$ contains an infinite-dimensional closed subspace where the ${l_{({p_n})}}$-topology and the ${\lambda _{({p_n})}}$-topology agree; (b) ${l_{({p_n})}}$ and ${\lambda _{({p_n})}}$ contain closed subspaces isomorphic to ${l_1}$; and (c) ${\lambda _{({p_n})}}$ contains no infinite-dimensional subspace where the ${\lambda _{({p_n})}}$-topology agrees with the ${l_1}$-topology if and only if \[ \lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .\]