Grothendieck space

In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space [math]\displaystyle{ X }[/math] in which every sequence in its continuous dual space [math]\displaystyle{ X^{\prime} }[/math] that converges in the weak-* topology [math]\displaystyle{ \sigma\left(X^{\prime}, X\right) }[/math] (also known as the topology of pointwise convergence) will also converge when [math]\displaystyle{ X^{\prime} }[/math] is endowed with [math]\displaystyle{ \sigma\left(X^{\prime}, X^{\prime \prime}\right), }[/math] which is the weak topology induced on [math]\displaystyle{ X^{\prime} }[/math] by its bidual. Said differently, a Grothendieck space is a Banach space for which a sequence in its dual space converges weak-* if and only if it converges weakly.

Characterizations

Let [math]\displaystyle{ X }[/math] be a Banach space. Then the following conditions are equivalent:

1. [math]\displaystyle{ X }[/math] is a Grothendieck space,
2. for every separable Banach space [math]\displaystyle{ Y, }[/math] every bounded linear operator from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ Y }[/math] is weakly compact, that is, the image of a bounded subset of [math]\displaystyle{ X }[/math] is a weakly compact subset of [math]\displaystyle{ Y. }[/math]
3. for every weakly compactly generated Banach space [math]\displaystyle{ Y, }[/math] every bounded linear operator from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ Y }[/math] is weakly compact.
4. every weak*-continuous function on the dual [math]\displaystyle{ X^{\prime} }[/math] 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 [math]\displaystyle{ X }[/math] must be reflexive, since the identity from [math]\displaystyle{ X \to X }[/math] is weakly compact in this case.
• Grothendieck spaces which are not reflexive include the space [math]\displaystyle{ C(K) }[/math] of all continuous functions on a Stonean compact space [math]\displaystyle{ K, }[/math] and the space [math]\displaystyle{ L^{\infty}(\mu) }[/math] for a positive measure [math]\displaystyle{ \mu }[/math] (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 [math]\displaystyle{ H^{\infty} }[/math] of bounded holomorphic functions on the disk is a Grothendieck space.

See also

• Dunford–Pettis property

References

• 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.
• Hazewinkel, Michiel, ed. (2001), "Grothendieck space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Khurana, Surjit Singh (1991). "Grothendieck spaces, II". Journal of Mathematical Analysis and Applications (Elsevier BV) 159 (1): 202–207. doi:10.1016/0022-247x(91)90230-w. ISSN 0022-247X. 
• Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.