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Transactions of the American Mathematical Society

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On the sequence space $ l\sb{(p\sb{n})}$ and $ \lambda \sb{(p\sb{n})},$ $ 0<p\sb{n}\leq 1$


Authors: S. A. Schonefeld and W. J. Stiles
Journal: Trans. Amer. Math. Soc. 225 (1977), 243-257
MSC: Primary 46A45; Secondary 46A15
DOI: https://doi.org/10.1090/S0002-9947-1977-0448027-X
MathSciNet review: 0448027
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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 {\vert{x_n}{\vert^{{p_n}}} < \infty } $, and let $ {\lambda _{({q_n})}}$ be the set of all real sequences $ ({y_n})$ such that $ {\sup _\pi }\sum {\vert{y_n}{\vert^{{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

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


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DOI: https://doi.org/10.1090/S0002-9947-1977-0448027-X
Keywords: Sequence spaces, nonlocally convex spaces, symmetric spaces, Schauder bases
Article copyright: © Copyright 1977 American Mathematical Society