# Grothendieck space

In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space X in which every weakly* convergent sequence in the dual space X* converges with respect to the weak topology of X*.

## Characterisations

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* 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(μ) for a positive measure μ (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 of bounded holomorphic functions on the disk is a Grothendieck space.[1]