A Banach space without a basis which has the bounded approximation property
The author proves the existence of separable Banach spaces X which have the bounded approximation property but do not have bases thus solving the last open problem concerning the relationship between the approximation property, the bounded approximation property and the existence of bases. After Enflo's and Pisier's this is the third method of constructing separable Banach spaces without a basis. The result actually is sharper and the spaces X have various additional properties: X is superreflexive, has an unconditional finite-dimensional decomposition but \(X\oplus B\) does not have local bases structure (i.e. there is no increasing sequence of finite-dimensional subspaces with dense union and uniformly bounded bases constants) for each B of cotype 2. In the real case, the construction can be modified in order to obtain spaces which do not admit complex structure and, consequently, are not isomorphic to the square of any Banach space. In the complex case (which has to be treated separately) X has a complex finite dimensional decomposition but not even a basis as a real space and X is not real- isomorphic to the square of any real Banach space. As a side product it turns out that with B being the \(\ell_ 2\)-sum of the \(\ell\) \(n_ 1\), the space \(Z=X\oplus B\) does not have local basis structure but its dual has local basis structure but certainly no basis.
constructing separable Banach spaces without a basis
unconditional finite-dimensional decomposition
scientific article; zbMATH DE number 4038489
scientific article