Generalized Köthe function spaces. I

The idea of constructing a space of functions taking values in a locally convex spaceEfrom a linear space of scalar valued functions is well known. We can, for example, define a space consisting of allE-valued functions φ(t) such that for all elements e′ of the dualE′ ofE. 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 spaceE. Therefore one usually starts from some function space which has interesting properties.