Computability-theoretic complexity of effective Banach spaces

We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.

We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.

However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.

We also investigate the effectivisations of certain classical theorems in Banachspaces.