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Uniqueness of unconditional bases of c 0 ( l p ) , 0 < p < 1

C. Leránoz — 1992

Studia Mathematica

We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of c 0 ( l p ) must be equivalent to a permutation of a subset of the canonical unit vector basis of c 0 ( l p ) . In particular, c 0 ( l p ) has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for c 0 ( l ₁ ) .

Uniqueness of unconditional basis of ℓ p ( c ₀ ) and ℓ p ( ℓ ₂ ) , 0 < p < 1

F. Albiac;  C. Leránoz — 2002

Studia Mathematica

We prove that the quasi-Banach spaces ℓ p ( c ₀ ) and ℓ p ( ℓ ₂ ) (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes ℓ ₁ ( c ₀ ) and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

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