Dual Spaces
Perhaps a better title for this chapter would be “Duality,” but this has a special meaning in functional analysis: an abstraction of the notion of a space X and its dual X ∗ into a pairing (X, Y ), where X is a vector space and Y is a space of linear func
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Dual Spaces
5.1 Adjoints Perhaps a better title for this chapter would be “Duality,” but this has a special meaning in functional analysis: an abstraction of the notion of a space X and its dual X into a pairing .X; Y /, where X is a vector space and Y is a space of linear functionals on X . The subject has its uses; the argument in Proposition 5.38 of Sect. 5.7 is based on such concepts. However, it leads away from the more practical functional analysis that is the subject of this book. Typically, one has some fairly specific spaces in mind, which then dictate the dual structure. Probably the most fundamental concept here that is not normally discussed in beginning graduate real analysis is the notion of an adjoint map (although the underlying idea often does appear in beginning graduate algebra!). Definition 5.1. Suppose X and Y are locally convex spaces, and T W X ! Y is a continuous linear map. The adjoint of T , denoted by T , is the map from Y to X defined by T .f / D f ı T . That is, T W Y ! X f 7! f ı T 8x 2 X; 8f 2 Y W ŒT .f /.x/ D f ŒT .x/: The preceding is a bit pedantic because the concept is slippery, appearing simultaneously trivial and mind-bending, particularly if one is interested in establishing results about T . Consider the following. Theorem 5.2. Suppose X and Y are locally convex spaces, and T W X ! Y is a continuous linear map. Letting X and Y denote their strong dual spaces, T W Y ! X is a continuous linear map. Also, if A X , then T .A/ı D .T /1 .Aı /. Finally, T is continuous when X and Y are equipped with their weak- topologies. M.S. Osborne, Locally Convex Spaces, Graduate Texts in Mathematics 269, 123 DOI 10.1007/978-3-319-02045-7__5, © Springer International Publishing Switzerland 2014
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5 Dual Spaces
Proof. Letting F denote the base field, if c 2 F, x 2 X , and f; g 2 Y , then T .cf /.x/ D .cf /.T .x// D cf .T .x// D cŒT .f /.x/ D ŒcT .f /.x/; and T .f C g/.x/ D .f C g/.T .x// D f .T .x// C g.T .x// D T .f /.x/ C T .g/.x/ D ŒT .f / C T .g/.x/; so T is linear. Next, suppose A X . Then for all f 2 Y : f 2 T .A/ı , 8x 2 A W jf .T .x//j 1 , 8x 2 A W jT .f /.x/j 1 , T .f / 2 Aı , f 2 .T /1 .Aı /: In particular, if F is finite in X , then .T /1 .F ı / D T .F /ı , so T is continuous when X and Y have their weak- topologies. Finally, T is strongly continuous since T is bounded: If A is bounded in X , so that Aı is a typical neighborhood of 0 in X , then .T /1 .Aı / D .T .A//ı is a strong neighborhood of 0 in Y . t u Now for the subtleties. The fact that T is linear is not a surprise, but note that it has nothing to do with the fact that T is linear. It simply follows from how F-valued functions are added together or multiplied by scalars. Also, the continuity proof made no use of the fact that T was continuous, only that T was bounded. What the (stronger) continuity condition does is guarantee that T takes values in X . If T were only bounded, then T .f / would be a bounded (bu
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