In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space X {\displaystyle X} in which every sequence in its continuous dual space X ′ {\displaystyle X^{\prime }} that converges in the weak-* topology σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} (also known as the topology of pointwise convergence) will also converge when X ′ {\displaystyle X^{\prime }} is endowed with σ ( X ′ , X ′ ′ ) , {\displaystyle \sigma \left(X^{\prime },X^{\prime \prime }\right),} which is the weak topology induced on X ′ {\displaystyle X^{\prime }} 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 X {\displaystyle X} be a Banach space. Then the following conditions are equivalent:
X {\displaystyle X} is a Grothendieck space, for every separable Banach space Y , {\displaystyle Y,} every bounded linear operator from X {\displaystyle X} to Y {\displaystyle Y} is weakly compact, that is, the image of a bounded subset of X {\displaystyle X} is a weakly compact subset of Y . {\displaystyle Y.}
for every weakly compactly generated Banach space Y , {\displaystyle Y,} every bounded linear operator from X {\displaystyle X} to Y {\displaystyle Y} is weakly compact. every weak*-continuous function on the dual X ′ {\displaystyle X^{\prime }} 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 X {\displaystyle X} must be reflexive, since the identity from X → X {\displaystyle X\to X} is weakly compact in this case. Grothendieck spaces which are not reflexive include the space C ( K ) {\displaystyle C(K)} of all continuous functions on a Stonean compact space K , {\displaystyle K,} and the space L ∞ ( μ ) {\displaystyle L^{\infty }(\mu )} for a positive measure μ {\displaystyle \mu } (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 H ∞ {\displaystyle H^{\infty }} 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. 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.