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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The Mackey topology and complemented subspaces of Lorentz sequence spaces $ d(w,p)$ for $ 0<p<1$


Authors: M. Nawrocki and A. Ortyński
Journal: Trans. Amer. Math. Soc. 287 (1985), 713-722
MSC: Primary 46A45; Secondary 46A10
DOI: https://doi.org/10.1090/S0002-9947-1985-0768736-7
MathSciNet review: 768736
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Abstract: In this paper we continue the study of Lorentz sequence spaces $ d(w,p)$, $ 0 < p < 1$, initiated by N. Popa [8]. First we show that the Mackey completion of $ d(w,p)$ is equal to $ d(v,1)$ for some sequence $ v$. Next, we prove that if $ d(w,p) \not\subset {l_1}$, then it contains a complemented subspace isomorphic to $ {l_p}$. Finally we show that if $ \lim {n^{ - 1}}\left(\sum\nolimits_{i = 1}^n {w_i}\right)^{1/p} = \infty $, then every complemented subspace of $ d(w,p)$ with symmetric bases is isomorphic to $ d(w,p)$.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768736-7
Keywords: $ p$-Banach spaces, Mackey topology, complemented subspaces
Article copyright: © Copyright 1985 American Mathematical Society