Compactness (or Heine-Borel compactness): Every open cover of A admits a finite subcover.
The Eberlein–Šmulian theorem states that the three are equivalent on a weak topology of a Banach space. While this equivalence is true in general for a metric space, the weak topology is not metrizable in infinite dimensional vector spaces, and so the Eberlein–Šmulian theorem is needed.
Applicationsedit
The Eberlein–Šmulian theorem is important in the theory of PDEs, and particularly in Sobolev spaces. Many Sobolev spaces are reflexive Banach spaces and therefore bounded subsets are weakly precompact by Alaoglu's theorem. Thus the theorem implies that bounded subsets are weakly sequentially precompact, and therefore from every bounded sequence of elements of that space it is possible to extract a subsequence which is weakly converging in the space. Since many PDEs only have solutions in the weak sense, this theorem is an important step in deciding which spaces of weak solutions to use in solving a PDE.
eberlein, Šmulian, theorem, mathematical, field, functional, analysis, named, after, william, frederick, eberlein, witold, lwowitsch, schmulian, result, that, relates, three, different, kinds, weak, compactness, banach, space, contents, statement, applications. In the mathematical field of functional analysis the Eberlein Smulian theorem named after William Frederick Eberlein and Witold Lwowitsch Schmulian is a result that relates three different kinds of weak compactness in a Banach space Contents 1 Statement 2 Applications 3 See also 4 References 5 BibliographyStatement editEberlein Smulian theorem 1 If X is a Banach space and A is a subset of X then the following statements are equivalent each sequence of elements of A has a subsequence that is weakly convergent in X each sequence of elements of A has a weak cluster point in X the weak closure of A is weakly compact A set A in any topological space can be compact in three different ways Sequential compactness Every sequence from A has a convergent subsequence whose limit is in A Limit point compactness Every infinite subset of A has a limit point in A Compactness or Heine Borel compactness Every open cover of A admits a finite subcover The Eberlein Smulian theorem states that the three are equivalent on a weak topology of a Banach space While this equivalence is true in general for a metric space the weak topology is not metrizable in infinite dimensional vector spaces and so the Eberlein Smulian theorem is needed Applications editThe Eberlein Smulian theorem is important in the theory of PDEs and particularly in Sobolev spaces Many Sobolev spaces are reflexive Banach spaces and therefore bounded subsets are weakly precompact by Alaoglu s theorem Thus the theorem implies that bounded subsets are weakly sequentially precompact and therefore from every bounded sequence of elements of that space it is possible to extract a subsequence which is weakly converging in the space Since many PDEs only have solutions in the weak sense this theorem is an important step in deciding which spaces of weak solutions to use in solving a PDE See also editBanach Alaoglu theorem Bishop Phelps theorem Mazur s lemma James theorem Goldstine theoremReferences edit Conway 1990 p 163 Bibliography editConway John B 1990 A Course in Functional Analysis Graduate Texts in Mathematics Vol 96 2nd ed New York Springer Verlag ISBN 978 0 387 97245 9 OCLC 21195908 Diestel Joseph 1984 Sequences and series in Banach spaces Springer Verlag ISBN 0 387 90859 5 Dunford N Schwartz J T 1958 Linear operators Part I Wiley Interscience Whitley R J 1967 An elementary proof of the Eberlein Smulian theorem Mathematische Annalen 172 2 116 118 doi 10 1007 BF01350091 S2CID 123175660 nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Eberlein Smulian theorem amp oldid 1188744913, wikipedia, wiki, book, books, library,
