Banach spaces without a local basis structure (Q1824150)

The paper contains solutions of some questions on local basis structure (LBS) posed by \textit{S. J. Szarek} [Banach spaces without bases which have the bounded approximation products, preprint (1985)]. (A separable Banach space is said to have the local basis structure iff it can be represented as a closure of an increasing sequence of finite dimensional subspaces having uniformly bounded basis constants.) The main results are: 1. There exists a Banach space with bounded approximation property (BAP) and with LBS but without basis. 2. There exists a Banach space with BAP but without LBS and without any non-trivial type. 3. For any \(q>2\) there exists a Banach space with BAP but without LBS and without cotype q. Reviewer's remark. The first and the second results are obtained also by \textit{S. J. Szarek} [Acta Math. 159, No.1-2, 81-98 (1987; Zbl 0637.46013)].