Generalized Köthe function spaces. I

Abstract

The idea of constructing a space of functions taking values in a locally convex space E from a linear space of scalar valued functions is well known. We can, for example, define a space consisting of all E -valued functions φ( t ) such that for all elements e′ of the dual E ′ of E . Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function space and the vector space E . Therefore one usually starts from some function space which has interesting properties.