On the sequence space $l_{(p_{n})}$ and $\lambda _{(p_{n})},$ $0<p_{n}\leq 1$
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- by S. A. Schonefeld and W. J. Stiles PDF
- Trans. Amer. Math. Soc. 225 (1977), 243-257 Request permission
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 \[ \lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .\]References
- Zvi Altshuler, P. G. Casazza, and Bor Luh Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973), 140–155. MR 328553, DOI 10.1007/BF02764600
- B. A. Barnes and A. K. Roy, Boundedness in certain topological linear spaces, Studia Math. 33 (1969), 147–156. MR 247402, DOI 10.4064/sm-33-2-147-156
- C. Bessaga, A note on universal Banach spaces of a finite dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958), 97–101. MR 0114108
- C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
- D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. (3) 16 (1966), 85–106. MR 192311, DOI 10.1112/plms/s3-16.1.85
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
- S. Rolewicz, On the characterization of Schwartz spaces by properties of the norm, Studia Math. 20 (1961), 87–92. MR 137986, DOI 10.4064/sm-20-1-87-92 S. Schonefeld and W. Stiles, On some linear topologies on $\cup {l_p}$ (submitted). —, On the inductive limit of $\cup {l_p},0 < p \leqslant 1$, Studia Math. (to appear).
- S. Simons, The sequence spaces $l(p_{\nu })$ and $m(p_{\nu })$, Proc. London Math. Soc. (3) 15 (1965), 422–436. MR 176325, DOI 10.1112/plms/s3-15.1.422
- W. J. Stiles, On non locally $p$-convex spaces, Colloq. Math. 23 (1971), 261–262. MR 310585, DOI 10.4064/cm-23-2-261-262
Additional Information
- © Copyright 1977 American Mathematical Society
- 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