Interplay between infinite-dimensional topology and functional analysis. Mappings defined by explicit formulas and their applications.

*(English)*Zbl 0840.46008Summary: We recall some explicit formulas of analytic character which were invented during the process of formation of infinite-dimensional topology, and present some applications of them. The following topics are covered:

A. Radial homeomorphisms and retractions of convex bodies; analogues of gauge functionals and radial retractions for Banach lattices. Applications: Lipschitz retraction onto \(c_0\) (Lindenstrauss) and lack of fixed points for Lipschitz self-maps of non compact convex sets (Lin-Sternfeld).

B. Non-complete-norm deleting homeomorphisms and diffeomorphisms with applications

(Garay) to ordinary differential equations. An analogy with West’s theorem on fixed point sets of transformation groups.

C. The coordinate switching technique: a “simultaneous” proof of West’s theorem and the Ribe-Aharoni-Lindenstrauss example of uniformly homeomorphic and not Lipschitz homeomorphic separable Banach spaces.

A. Radial homeomorphisms and retractions of convex bodies; analogues of gauge functionals and radial retractions for Banach lattices. Applications: Lipschitz retraction onto \(c_0\) (Lindenstrauss) and lack of fixed points for Lipschitz self-maps of non compact convex sets (Lin-Sternfeld).

B. Non-complete-norm deleting homeomorphisms and diffeomorphisms with applications

(Garay) to ordinary differential equations. An analogy with West’s theorem on fixed point sets of transformation groups.

C. The coordinate switching technique: a “simultaneous” proof of West’s theorem and the Ribe-Aharoni-Lindenstrauss example of uniformly homeomorphic and not Lipschitz homeomorphic separable Banach spaces.

##### MSC:

46B20 | Geometry and structure of normed linear spaces |

46B42 | Banach lattices |

58C30 | Fixed-point theorems on manifolds |

52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |