A~theorem on projections of rearranged series with terms in~$L_p$

D. V. Pecherskii

Izvestiya. Mathematics , Tome 11 (1977) no. 1, pp. 193-204

Résumé

The following theorem is proved in this paper: if a series $\sum_{k=1}^\infty f_k$ with terms in $L_p$ ($1\leqslant p\infty$) satisfies either the condition $\sum_{k=1}^\infty\|f_k\|^2\infty$ when $2\leqslant p\infty$ or the condition $\sqrt{\sum_{k=1}^\infty f_k^2(x)}\in L_p$ when $1\leqslant p2$, then in order that there exist a permutation of the natural numbers $\{n_1,\dots,n_k,\dots\}$ such that $\sum_{k=1}^\infty f_{n_k}=f$ in the $L_p$ norm, it is necessary and sufficient that for each linear functional $F\in L_p^*$, $\|F\|=1$, there exists a permutation $\{m_1,\dots,m_k,\dots\}$ depending on $F$ such that $\sum_{k=1}^\infty F(f_{m_k})=F(f)$. Bibliography: 9 titles.