Preliminaries

Abstract

We recommend to browse the material of this chapter even if the reader is acquainted with the concepts listed in the title. A beginner finds a self-contained treatment mostly with full proofs and only few references to text books. The operators of integral equations examined in this book considered between suitably spaces belong to the class of so called Fredholm (or Noether) operators. We begin the chapter with classical results about the existence and uniqueness of a solution to an equation of the form x = Tx + f where T ∈ L(χ) is a compact operator in a Banach space χ, f ∈ χ is given and x ∈ χ is looked for. These results have straightforward counterpart for the equations of the type Ax := Bx + Cx = y where B ∈ L (χ,Y) is an invertible and C ∈ L(χ,Y) is a compact operator between Banach spaces χ and Y. Such operators A ∈ L(χ, Y) cover the class of Fredholm operators of index 0 but we consider also the operators of arbitrary index. During the present chapter we use the special notations χ and Y for Banach spaces whereas X and Y refer to the vector spaces. We widely use the generalized concept of dual operators by Jörgens, cf. [Jör82],[Kre89a]. Some of results seem to be novel at least methodically.KeywordsBanach SpaceBilinear FormCompact OperatorClosed SubspaceFredholm OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.