This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-requisites and get a feeling about this area of Banach space theory. Many open problems of different level of difficulty are discussed. For the reader...

We survey the recent investigations on approximate amenability/contractibility and pseudo-amenability/contractibility for Banach algebras. We will discuss the core problems concerning these notions and address the significance of any solutions to them to the development of the field. A few new results are also included.

We survey some recent developments in the theory of Fréchet spaces and of their duals. Among other things, Section 4 contains new, direct proofs of properties of, and results on, Fréchet spaces with the density condition, and Section 5 gives an account of the modern theory of general Köthe echelon and co-echelon spaces. The final section is devoted to the developments in tensor products of Fréchet spaces since the negative solution of Grothendieck?s ?problème des topologies?.

Geometric structure of Cesàro function spaces $Ce{s}_p\left(I\right)$, where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $Ce{s}_p[0,1]$ contains isomorphic and complemented copies of $l_q$-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $Ce{s}_p[0,1]$.

On étudie les convexes compacts $K$, tels que pour toute partie $A$ de $K$, l’ensemble des fonctions affines continues sur $K$, comprises entre 0 et 1, et nulles sur $A$, ait un plus grand élément. On caractérise ces convexes compacts comme ceux dont des quotients affines convenables sont des chapeaux universels de cônes à base compacte. On a une “complémentation naturelle” sur le treillis des faces exposés de $K$, et des liens remarquables entre ce treillis et l’espace des fonctions affines continues sur $K$.