A local version of a result of Gamlen and Gaudet
Let D be a subset of the natural numbers and \(X^ p_ D\) the closed linear span in \(L^ P[0,1]\) of the subset \(\{h_ j| j\in D\}\) of the Haar basis \(\{h_ j\}^{\infty}_{j=1}\) for \(L^ p[0,1]\), where \(1<p<+\infty.\) \textit{J. L. B. Gamlen} and \textit{R. J. Gaudet} showed in 1973 that if D is infinite then \(X^ p_ D\) is isomorphic to either \(\ell^ p\) or \(L^ p[0,1]\) [Isr. J. Math. 15, 404-413 (1973; Zbl 0296.46031)]. In this paper it is shown that for any \(1<p<+\infty\) there exists a constant \(K_ p>0\) so that for any finite set D the space \(X^ p_ D\) is \(K_ p\)- isomorphic to \(\ell^ p_{\#D}\), where {\#}D denotes the cardinality of D.
scientific article; zbMATH DE number 4080147