On a Theorem of Kadets and Pełczyński

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 { u n } n = 1 ∞ is an unconditional basic sequence in X such that ‖ u n ‖ X ≍ ‖ u n ‖ L 1 , n ∈ ℕ , then the norms of the spaces X and L 1 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.