Locally Convex Spaces by M. Scott Osborne

Author:M. Scott Osborne
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

There is one situation where a separately continuous bilinear map arises naturally, and it illustrates the limits to generalizing Corollary 4.18: the evaluation map. Suppose Y is a Hausdorff locally convex space, and X = Y ∗, its strong dual. Let be the base field, and define by e(f, y) = f(y). e(f, ?) = f, which is continuous; while e(?, y) has values of modulus on , and so is also continuous. That is, the evaluation map is always separately continuous. However, it is not jointly continuous unless Y can be normed; see Exercise 5. If Y is an LB-space, then Y ∗ is a Fréchet space (Proposition 3.46), and so is first countable, while Y itself is barreled (Corollary 4.6). This shows that something beyond “X is first countable and Y is barreled” is necessary for Corollary 4.18. Incidentally, it also shows that the strong dual of a Fréchet space Y cannot be another Fréchet space unless Y can actually be normed. It cannot even be first countable …but (thanks to results discussed in the next section) the strong dual of a Fréchet space is always complete.

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