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
editLet 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
edit- 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.[1]
See also
editReferences
edit- ^ J. Bourgain, 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
- Khurana, Surjit Singh (1991). "Grothendieck spaces, II". Journal of Mathematical Analysis and Applications. 159 (1). Elsevier BV: 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.