Grothendieck space

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

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.
4. every weak*-continuous function on the dual X* is weakly Riemann integrable.

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 space H^∞ of bounded holomorphic functions on the disk is a Grothendieck space.[1]

See also

• Dunford–Pettis property

References

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) [1994], "Grothendieck space", Encyclopedia of Mathematics, EMS Press
• Nisar A. Lone, on weak Riemann integrablity of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.