Generalized Köthe function spaces. I
Published online by Cambridge University Press: 24 October 2008
Extract
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.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 3 , May 1969 , pp. 601 - 611
- Copyright
- Copyright © Cambridge Philosophical Society 1969
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