Some problems in the Banach space theory (Q1104515)

Fourteen theorems with sketches of proofs are presented about linear- topological properties of a Banach space \(X_ 1\) which admits dense and weakly compact embedding into some Banach space X (shortly, \(X_ 1\in E(X))\). For example: Theorem 3. \(X_ 1\) is reflexive if and only if every of its subspaces of codimension one is isomorphic to a dual space. Theorem 9. Separable \(X_ 1\) has separable dual if and only if \(X_ 1\) contains no subspaces isomorphic to \(\ell_ 1.\) Theorem 10. \(X_ 1\) is reflexive if and only if \(X_ 1\) contains no subspaces isomorphic to \(c_ 0\) or \(\ell_ 1.\) Reviewer's remark. Counterexamples to this theorems are respectively: \(X_ 1=\ell_ 1\), \(X_ 1=JT\) (James tree), \(X_ 1=J\) (James quasi- reflexive space).