Grothendieck space

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In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space X every weakly* convergent sequence in the dual space X* converges with respect to the weak topology of X*.

Characterisations[edit]

Let X be a Banach space. Then the following conditions are equivalent:

  1. X is a Grothendieck space,
  2. for every separable Banach space Y, every bounded linear operator from X to Y is weakly compact, that is, the image of a bounded subset of X is a weakly compact subset of Y,
  3. for every weakly compactly generated Banach space Y, every bounded linear operator from X to Y is weakly compact.
  1. X is Grothendieck iff every weak* - continuous function on the dual is weakly - Riemann integrable.

Examples[edit]

  • Every reflexive Banach space is a Grothendieck space. Conversely, it is a consequence of the Eberlein–Šmulian theorem that a separable Grothendieck space X must be reflexive, since the identity from X to X is weakly compact in this case.
  • Grothendieck spaces which are not reflexive include the space C(K) of all continuous functions on a Stonean compact space K, and the space L(μ) for a positive measure μ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).
  • Jean Bourgain proved that the disc algebra H is a Grothendieck space.[1]

See also[edit]

References[edit]

  1. ^ J. Bourgain, H is a Grothendieck space, Studia Math., 75 (1983), 193–216.
  • J. Diestel, Geometry of Banach spaces, Selected Topics, Springer, 1975.
  • J. Diestel, J. J. Uhl: Vector measures. Providence, R.I.: American Mathematical Society, 1977. ISBN 978-0-8218-1515-1.
  • Shaw, S.-Y. (2001), "G/g110250", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  • Nisar A. Lone, on weak Riemann integrablity of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.