Kerin-Šmulian's theorem

The theorem of Kerin-Šmulian , named after Mark Grigorjewitsch Kerin and Witold Lwowitsch Šmulian , is a mathematical theorem from functional analysis that represents a criterion for the closure of a convex set with respect to the weak - * - topology .

Formulation of the sentence

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 : ${\displaystyle E}$ ${\displaystyle E_{r}'}$ ${\displaystyle r}$ ${\displaystyle E}$ ${\displaystyle r>0}$ ${\displaystyle M\subset E'}$ ${\displaystyle M\cap E_{r}',\,r>0}$ ${\displaystyle M}$

• Kerin-Šmulian theorem : Let be a Banach space and a convex set. If every weak - * - is closed, then weak - * - is closed too . ${\displaystyle E}$ ${\displaystyle M\subset E'}$ ${\displaystyle M\cap E_{r}'}$ ${\displaystyle r>0}$ ${\displaystyle M}$

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 . ${\displaystyle M}$ ${\displaystyle F_{n}\subset E^{'}}$ ${\displaystyle n}$ ${\displaystyle F_{1}\subset F_{2}\subset F_{3}\subset \ldots }$ ${\displaystyle S_{n}:=\{f\in F_{n};\|f\|=n\}}$ ${\displaystyle n}$ ${\displaystyle F_{n}}$ ${\displaystyle M_{n}\subset S_{n}}$ ${\displaystyle M=M_{1}\cup M_{2}\cup M_{3}\cup \ldots .}$

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 . ${\displaystyle M\cap E_{r}'}$ ${\displaystyle r>0}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle U=\{f\in E\,';|f(x_{1})|<\epsilon ,\ldots |f(x_{m})|<\epsilon \}}$ ${\displaystyle x_{1},\ldots ,x_{m}\in E}$ ${\displaystyle \epsilon >0}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle \max _{i=1,\ldots m}\|x_{i}\|<n\epsilon }$ ${\displaystyle n>m}$ ${\displaystyle g\in S_{n}}$ ${\displaystyle g(x_{1})=\ldots =g(x_{m})=0}$ ${\displaystyle f\in M_{n}}$ ${\displaystyle \|f-g\|<1/n}$ ${\displaystyle f\in M\cap U}$ ${\displaystyle |f(x_{i})|=|f(x_{i})-g(x_{i})|\leq \|f-g\|\cdot \|x_{i}\|<\epsilon }$ ${\displaystyle i=1,\ldots ,m}$

The bw * topology

→ Main article : Constrained weak - * topology

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: ${\displaystyle M\subset E'}$ ${\displaystyle M\cap E_{r}'}$ ${\displaystyle r>0}$

• Let be a Banach space and a convex set. Then the weak - * - degree and the bw * - degree of coincide. ${\displaystyle E}$ ${\displaystyle M\subset E'}$ ${\displaystyle M}$

Banach-Dieudonné theorem

• Let be a Banach space and a subspace. is weak - * - completed if and only if weak - * - is completed. ${\displaystyle E}$ ${\displaystyle U\subset E'}$ ${\displaystyle U}$ ${\displaystyle U\cap E_{1}^{'}}$

This sentence, named after Banach and Dieudonné , is apparently a corollary to the sentence of Kerin-Šmulian. ${\displaystyle U\cap E_{r}^{'}=r\cdot (U\cap E_{1}^{'})}$

swell

• MM Day: Normed Linear Spaces Springer-Verlag GmbH, third edition (1973) ISBN 3540061487