On Tensor Products of Quasi-Banach Spaces

Abstract

Due to applications in approximation theory we are interested in tensor products of quasi-Banach spaces. Though a general abstract theory seems not possible beyond basic topological issues because the dual spaces are possibly trivial, we aim at extending some basic notions like crossnorms, reasonable and uniform norms. In the present paper this is done for quasi- Banach spaces with separating duals, and this condition turns out to be the (in a certain sense) minimal requirement. Moreover, we study extensions of the classical injective and p-nuclear tensor norms to quasi Banach spaces. In particular, we give a sufficient condition for the $p$nuclear quasi-norms to be crossnorms, which particularly applies to the case of weighted $l_p$-sequence spaces.