The Pontrjagin Duality Theorem in Linear Spaces

Abstract

The Pontrjagin Duality Theorem, known to be true for locally compact groups, asserts that the given group and the character group of its character group are isomorphic under a natural mapping. So-called linear topological spaces (l.t.s.) are equivalent to their second conjugate spaces, again under a natural mapping. This raises two questions: first, for what class of topological groups does the Pontrjagin theorem hold? Second, what relation, if any, is there between the two kinds of theorem when considered in linear spaces? The present paper answers the first question in part,. bly showing that in an extensive class of infinite-dimensional (and thus not locally compact) topological linear spaces the Pontrjagin theorem is valid, and the second by showing that in reflexive linear topological spaces the group-duality theorem is also true and, in fact, merely restates the reflexive property. The key observation which makes our results possible is very simple: if f is a linear functional on a real l.t.s. then the function x defined by: x(x) = exp (if(x)) for each x in the space, is a character. The author's proof of the main theorem as presented to the Society (abstract 209t, Dec. '48) leaned heavily on Arens' paper [1].' It has since been possible to make this investigation more self-contained, and in so doing, to generalize its principal result from holding for real reflexive Banach spaces to arbitrary locally convex real l.t.s. which are either reflexive (though not necessarily either complete or normable) or Banach spaces (though not necessarily reflexive). This has served also to settle the converse problem: since completeness and normability suffice to make the Pontrjagin theorem valid it becomes clear that reflexivity is not a necessary condition for, and so certainly not identical with, the group-duality theorem, though in the special case of reflexive spaces the two sorts of duality coincide. It should be pointed out that despite the fact that Arens' results are not too obviously or heavily leaned upon in this paper in its present form, the ideas in [1] supplied the clue and motive, as well as occasionally the method for the discovery of these theorems. The author is indebted also to J. L. Kelley and to Irving Kaplansky for helpful discussions, and to the latter especially for the essential ideas in the proof of Lemma 1.