Nuclear spaces were discovered by A. Grothendieck, and the most important part of the theory of nuclear spaces was developed in his famous article (see Grothendieck). Unfortunately, the machinery of tensor products used there is very cumbersome; this makes the theory some unnecessary difficul ties and complicacy. Pietsch was the first to simplify the theory of nuclear spaces by using locally convex spaces of summable and absolutely summable families, instead of locally convex tensor products. It is my impression that the idea of using seminorms satisfying some expected properties to investigate some classes of locally convex spaces, would make the development of the whole theory easier. This has been done by the author for stuing the topology 0' S of uniform convergence on orderwbounded sets. Thus, one of the purposes of these lecture notes is to embark upon a unifying treatment of Schwartz spaces, nuclear spaces, and A-nuclear spaces by using this idea. Therefore, this monograph is actually a continuation of that written by the author.
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