Grothendieck space
# 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 $\sigma \left(X^{\prime },X\right)$ (also known as the topology of pointwise convergence) will also converge when $X^{\prime }$ is endowed with $\sigma \left(X^{\prime },X^{\prime \prime }\right),$ 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:

* $X$ is a Grothendieck space,
* 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 $Y.$
* for every weakly compactly generated Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact.
* 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 $L^{\infty }(\mu )$ 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 $H^{\infty }$ of bounded holomorphic functions on the disk is a Grothendieck space.
