In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space in which every sequence in its continuous dual space that converges in the weak-* topology (also known as the topology of pointwise convergence) will also converge when is endowed with which is the weak topology induced on 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 be a Banach space. Then the following conditions are equivalent: is a Grothendieck space, for every separable Banach space every bounded linear operator from to is weakly compact, that is, the image of a bounded subset of is a weakly compact subset of for every weakly compactly generated Banach space every bounded linear operator from to is weakly compact. every weak*-continuous function on the dual 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 must be reflexive, since the identity from is weakly compact in this case. Grothendieck spaces which are not reflexive include the space of all continuous functions on a Stonean compact space and the space for a positive measure (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 of bounded holomorphic functions on the disk is a Grothendieck space.J. Bourgain, is a Grothendieck space, Studia Math., 75 (1983), 193–216.
See also
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. .
Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.