On a Theorem of Kadets and Pełczyński

Resumo

Necessary and sufficient conditions are found under which a symmetric space X on [0,1] of type 2 has the following property, which was first proved for the spaces L_p, p > 2, by Kadets and Pełczyński: if \(\left\{ {{u_n}} \right\}_{n = 1}^\infty \) is an unconditional basic sequence in X such that

\({\left\| {{u_n}} \right\|_X}\;\asymp\;{\left\| {{u_n}} \right\|_{{L_1}}},\;\;\;\;\;\;\;\;n\; \in \;\mathbb{N},\)

then the norms of the spaces X and L₁ are equivalent on the closed linear span [u_n] in X. For sequences of martingale differences, this implication holds in any symmetric space of type 2.