Grothendieck space
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 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 .
- Jean Bourgain proved that the space of bounded holomorphic functions on the disk is a Grothendieck space.