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Grantor: Czech Science Foundation (GAČR)

**Objectives**:

The subject of our proposal are abstract problems concerning nonlinear mappings between Banach spaces and their subsets. For easier orientation, it is convenient to divide the project into the following five interdependent areas.

1. General properties of uniform mappings, their reduction to Lipschitz mappings, and their metric properties.

2. Linearization properties of Lipschitz mappings, in particular the existence of derivatives.

3. Structural properties of participating spaces, linear theory.

4. Renormings of Banach spaces.

5. Applications to other areas of mathematics, such as fixed point theory and differential equations.

Concrete examples of the proposed problems:

Are the classical Banach function spaces linearly isomorphic to their uniformly homeomorphic images?

Is the unit ball uniformly homeomorphic to the unit sphere?

Are Lipschitz isomorphic separable Banach spaces linearly isomorphic?

What are the complemented subspaces of the classical function Banach spaces?

Do reflexive Banach spaces have a fixed point property for nonexpansive mappings?

Main investigator:

Participating institutions:

Institute of Mathematics, Czech Academy of Sciences, **Coordinator**

Faculty of Electrical Engineering, Czech Technical University in Prague

Faculty of Mathematics and Physics, Charles University in Prague