Construction of an orthonormal basis in $C^{m}(I^{d})$ and $W^{m}_{p}(I^{d})$

Z. Ciesielski; J. Domsta

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Uniqueness of unconditional bases in $c_{0}$ -products

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We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_{0} (X)$ . We also give some positive results including a simpler proof that $c_{0} (ℓ_{1})$ has a unique unconditional basis and a proof that $c_{0} (ℓ_{p_{n}}^{N_{n}})$ has a unique unconditional basis when $p_{n} ￬ 1$ , $N_{n + 1} \geq 2 N_{n}$ and $(p_{n} - p_{n + 1}) l o g N_{n}$ remains bounded.

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Uniqueness of unconditional bases of $c_{0} (l_{p})$ , 0 < p < 1

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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 $c_{0} (l ₁)$ .

Displaying similar documents to “Construction of an orthonormal basis in C m ( I d ) and W p m ( I d ) ”

Displaying similar documents to “Construction of an orthonormal basis in $C^{m} (I^{d})$ and $W_{p}^{m} (I^{d})$ ”