Bases in spaces of multilinear forms over Banach spaces
Let \(E_1,\dots,E_m\) be Banach spaces with Schauder bases \((e^1_n),\dots,(e^m_n)\) and let \({\mathcal L}^m(E_1\times\cdots\times E_m)\) be the space of all continuous, \(m\)-linear, complex-valued functionals over \(E_1\times\cdots\times E_m\). There are investigated properties equivalent to the relation \(|A_n|\to 0\) for \(A\in{\mathcal L}^m(E_1\times\cdots\times E_m)\). As corollaries it is proved that 1) if \(E_1,\dots,E_m\) are unconditional shrinking bases, then either monomials form a basis of \({\mathcal L}^m(E_1\times\cdots\times E_m)\) or \({\mathcal L}^m(E_1\times\cdots\times E_m)\) contains a copy of \(\ell^\infty\), 2) if \(E_1,\dots,E_m\) are reflexive and have Schauder bases, then \({\mathcal L}^m(E_1\times\cdots\times E_m)\) is reflexive if and only if it has a monomial basis. These results are applied to obtain some estimations for lower and upper indices in the sense of R. Gonzalo and J. A. Jaramillo of spaces \(E_1,\dots,E_m\) and their duals \(E^*_j\). There are shown examples of sequence spaces \(E_j=\ell^p\), Orlicz sequence spaces and others.
space of all continuous, \(m\)-linear, complex-valued functionals
scientific article; zbMATH DE number 908591