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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*.

Contents

Characterisations

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.

Examples

  • 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.

    • References

    Grothendieck space Wikipedia
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