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.[1]
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. 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.