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The Mackey topology and complemented subspaces of Lorentz sequence spaces d(w,p) for \(0<p<1\). (English) Zbl 0537.46012

In this paper we continue the study of Lorentz sequence spaces d(w,p), \(0<p<1\), initiated by N. Popa [Trans. Am. Math. Soc. 263, 431-456 (1981; Zbl 0461.46006)]. 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 \(d(w,p)\not\subset \ell_ 1\), then it contains a complemented subspace isomorphic to \(\ell_ p\). Finally, we show that if \(\lim_{n\to \infty}n^{-1}(\sum^{n}_{i=1}w_ i)^{1/p}=\infty,\) then every complemented subspace of d(w,p) with symmetric basis is isomorphic to d(w,p).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46A35 Summability and bases in topological vector spaces
46B25 Classical Banach spaces in the general theory
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)

Citations:

Zbl 0461.46006
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