Inmathematics, aGrothendieck space, named afterAlexander Grothendieck, is aBanach space${\displaystyle X}$ in which every sequence in itscontinuous dual space${\displaystyle X^{\mathrm{\prime }}}$ that converges in theweak-* topology${\displaystyle \sigma \left(X^{\mathrm{\prime }},X\right)}$ (also known as thetopology of pointwise convergence) will also converge when${\displaystyle X^{\mathrm{\prime }}}$ is endowed with${\displaystyle \sigma \left(X^{\mathrm{\prime }},X^{\mathrm{\prime }\mathrm{\prime }}\right),}$ which is theweak topology induced on${\displaystyle X^{\mathrm{\prime }}}$ by itsbidual. Said differently, a Grothendieck space is a Banach space for which asequence in its dual space converges weak-* if and only if it converges weakly.

Let${\displaystyle X}$  be a Banach space. Then the following conditions are equivalent:

1. ${\displaystyle X}$  is a Grothendieck space,
2. for everyseparable Banach space${\displaystyle Y,}$  everybounded linear operator from${\displaystyle X}$  to${\displaystyle Y}$  isweakly compact, that is, the image of a bounded subset of${\displaystyle X}$  is a weakly compact subset of${\displaystyle Y.}$ 
3. for every weakly compactly generated Banach space${\displaystyle Y,}$  every bounded linear operator from${\displaystyle X}$  to${\displaystyle Y}$  isweakly compact.
4. every weak*-continuous function on the dual${\displaystyle X^{\mathrm{\prime }}}$  is weakly Riemann integrable.

• Everyreflexive Banach space is a Grothendieck space. Conversely, it is a consequence of theEberlein–Šmulian theorem that a separable Grothendieck space${\displaystyle X}$  must be reflexive, since the identity from${\displaystyle X\to X}$  is weakly compact in this case.
• Grothendieck spaces which are not reflexive include the space${\displaystyle C(K)}$  of all continuous functions on aStonean compact space${\displaystyle K,}$  and the space${\displaystyle L^{\mathrm{\infty }}(\mu )}$  for apositive measure${\displaystyle \mu }$  (a Stonean compact space is aHausdorff compact space in which theclosure of everyopen set is open).
• Jean Bourgain proved that the space${\displaystyle H^{\mathrm{\infty }}}$  of bounded holomorphic functions on the disk is a Grothendieck space.^[1]

• Dunford–Pettis property

• 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.