Geometry of some non-classical Banach spaces.

The project focuses on open research questions connected to the geometry of non-classial Banach spaces of type Müntz and Schreier. Some of the questions are intimately linked to a modern and active research direction in Banach space theory where different geometric properties of Banach spaces called diameter 2 properties, are considered. The reason for this is that diameter 2 properties and octahedrality properties are often dual properties in the sense that a Banach space has a diameter 2 property if and only the dual has an octahedrality property. Other questions are connected to extreme points in the unit ball of non-classical Müntz spaces, and one question is does in the direction of non-linear embeddings of Schreier spaces into Banach spaces. Together with the French partner team we want to explore the following questions: 1. Characterize the extreme points of the unit ball of the dual of a Müntz space. 2. Characterize the extreme points of the unit ball of a Müntz space. 3. Investigate when the dual of a Müntz space is L-embedded. 4. Investigate when Müntz spaces X as subspaces of the space of integrable functons on the unit interval are octahedral. 5. Characterize the extreme points of the unit ball of the Müntz spaces X in Question 4. 6. Does Lipschitz embedding with distortion < 2 of Schreier spaces X_(S_? ) into a Banach space X provide a corresponding lower bound in the Szlenk index of X?