On some generalizations of bases in Banach spaces

This article considers pseudobases and quasibases in Banach spaces, as introduced by Gelbaum. A geometric characterization of pseudobases is established. It is proved that pseudobases are stable. It is shown that pseudobases and quasibases in the $L^p$-spaces do not, in general, have an interpolation property with respect to these spaces which is inherent to bases. Namely, an example is constructed of a system of functions that is an unconditional quasibasis in $L^2(0,\,1)$ and $L^q(0,\,1)$ ($q\in(1,\,2)$ fixed) and at the same time is not a pseudobasis in any $L^p(0,\,1)$ with $p\in(q,\,2)$ for any rearrangement of it.
Bibliography: 10 titles.