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On a problem posed by M. M. Popov

F. Albiac; J. L. Ansorena — 2012

Studia Mathematica

We show that if X is a non-locally convex quasi-Banach space with a rich dual, there exists a continuous function f: [0,1] → X failing to have a primitive. This answers a twenty year-old question raised by M. Popov in this journal.

Conditionality constants of quasi-greedy bases in super-reflexive Banach spaces

F. Albiac; J. L. Ansorena; G. Garrigós; E. Hernández; M. Raja — 2015

Studia Mathematica

We show that in a super-reflexive Banach space, the conditionality constants $k_{N} (ℬ)$ of a quasi-greedy basis ℬ grow at most like $O ({(l o g N)}^{1 - ε})$ for some 0 < ε < 1. This extends results by the third-named author and Wojtaszczyk (2014), where this property was shown for quasi-greedy bases in $L_{p}$ for 1 < p < ∞. We also give an example of a quasi-greedy basis ℬ in a reflexive Banach space with $k_{N} (ℬ) \approx l o g N$ .

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