With a few exceptions the locally convex spaces encountered in analysis can be divided into two classes. First, there are the normed spaces, which belong to classical functional analysis, and whose theory can be considered essentially closed. The second class consists of the so-called nuclear locally convex spaces, which were introduced in 1951 by A. Grothendieck. The two classes have a trivial intersection, since it can be shown that only finite dimensional locally convex spaces are simultaneously normable and nuclear.
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