Bilinear mappings – selected properties and problems

Abstract

This work is a survey of results and problems connected with selected properties of bilinear mappings between function spaces. Section 1 is devoted to the lack of the counterpart of the Banach openness principle for bilinear surjections between Banach spaces. Some counterexamples and positive results are discussed. Then we deal with variants of openness for multiplication in spaces C(K), for a compact K, and in other Banach algebras and function spaces. In Section 2, several phenomena of dichotomies for operators of multiplication and convolution in spaces of integrable or continuous functions are presented. Some natural properties of these operators hold either always or the sets of objects having the given property are small. The smallness is described by meagerness and porosity. Let us introduce some basic notation. The ball with center x and radius r in a given metric space X is denoted by B X (x, r) (if X is fixed, index X will be omitted). The interior of a set A ⊂ X will be written as int(A).