In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space in which every sequence in its continuous dual space that converges in the weak-* topology (also known as the topology of pointwise convergence which is the weak topology 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

be a Banach space. Then the following conditions are equivalent:

1. is a Grothendieck space,

2. for every separable every bounded linear operator is weakly compact

3. is weakly compact.

4. is weakly Riemann integrable.

Examples

• Every reflexive Banach space is a Grothendieck space. Conversely, it is a consequence of the Eberlein–Šmulian theorem is weakly compact in this case.

• of all continuous functions on a Stonean for a positive measure (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).

• Jean Bourgain of bounded holomorphic functions on the disk is a Grothendieck space.

See also

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. .

• Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.