If a Banach space , then let the closed
sphere be in the dual space of , where let. According to the Banach-Alaoglu theorem, this is compact with regard to the weak - * - topology and therefore closed. So if a weakly - * - closed subset, then the sets are weakly - * - closed. The theorem to be discussed here states that the converse also holds for convex sets :
Kerin-Šmulian theorem : Let be a Banach space and a convex set. If every weak - * - is closed, then weak - * - is closed too .
Remarks
An example
As the following example shows, the statement of the Kerin-Šmulian theorem is false if it is not convex. For this, let -dimensional subspaces with and let the spherical surface with radius be in . Since these spherical surfaces are compact, there is a finite 1 / n network . Set .
Then for each one is finite and therefore weak - * - closed. but itself is not weak - * - closed, because 0 is in the weak - * - closure of . For this purpose it has to be shown that every set of the form , where and , contains an element from . Choose so big that and . Because of the latter, there is a with for dimensional reasons . Now choose one with . Then is because for everyone .
Declare a set complete when the average for each weak - * - is closed. It is easy to consider that this defines a topology , the so-called bw * topology . As the above example shows, in the case of infinite-dimensional Banach spaces this topology is really finer than the weak - * - topology. Kerin-Šmulian's theorem can now be reformulated as follows:
Let be a Banach space and a convex set. Then the weak - * - degree and the bw * - degree of coincide.
Banach-Dieudonné theorem
Let be a Banach space and a subspace. is weak - * - completed if and only if weak - * - is completed.
This sentence, named after Banach and Dieudonné , is apparently a corollary to the sentence of Kerin-Šmulian.
swell
MM Day: Normed Linear Spaces Springer-Verlag GmbH, third edition (1973) ISBN 3540061487