Grothendieck space

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

1. $X$ is a Grothendieck space,
2. for every separable Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact, that is, the image of a bounded subset of $X$ is a weakly compact subset of $Y.$
3. for every weakly compactly generated Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact.
4. every weak*-continuous function on the dual $X^{\prime }$ 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 $X$ must be reflexive, since the identity from $X\to X$ is weakly compact in this case.
• Grothendieck spaces which are not reflexive include the space $C(K)$ of all continuous functions on a Stonean compact space $K,$ and the space $L^{\mathrm{\infty }}(\mu )$ for a positive measure $\mu$ (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 $H^{\mathrm{\infty }}$ 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.
• Hazewinkel, Michiel, ed. (2001), "Grothendieck space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Khurana, Surjit Singh (1991). "Grothendieck spaces, II". Journal of Mathematical Analysis and Applications (Elsevier BV) 159 (1): 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.