Grothendieck space

In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space ${\displaystyle X}$ in which every sequence in its continuous dual space ${\displaystyle X^{\prime }}$ that converges in the weak-* topology ${\displaystyle \sigma \left(X^{\prime },X\right)}$ (also known as the topology of pointwise convergence) will also converge when ${\displaystyle X^{\prime }}$ is endowed with ${\displaystyle \sigma \left(X^{\prime },X^{\prime \prime }\right),}$ which is the weak topology induced on ${\displaystyle X^{\prime }}$ 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 ${\displaystyle X}$ be a Banach space. Then the following conditions are equivalent:

1. ${\displaystyle X}$ is a Grothendieck space,
2. for every separable Banach space ${\displaystyle Y,}$ every bounded linear operator from ${\displaystyle X}$ to ${\displaystyle Y}$ is weakly 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 ${\displaystyle Y}$ is weakly compact.
4. every weak*-continuous function on the dual ${\displaystyle X^{\prime }}$ 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 ${\displaystyle X}$ 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 a Stonean compact space ${\displaystyle K,}$ and the space ${\displaystyle L^{\infty }(\mu )}$ for a positive measure ${\displaystyle \mu }$ (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 ${\displaystyle H^{\infty }}$ of bounded holomorphic functions on the disk is a Grothendieck space.^[1]

See also

• Dunford–Pettis property

References

1. J. Bourgain, ${\displaystyle H^{\infty }}$ 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.