Grothendieck space explained

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} \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,

1. 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.

1. for every weakly compactly generated Banach space

Y, every bounded linear operator from X to Y is weakly compact.

1. 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.^[1]

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. .
  • Khurana. Surjit Singh. Grothendieck spaces, II. Journal of Mathematical Analysis and Applications. Elsevier BV. 159. 1. 1991. 0022-247X. 10.1016/0022-247x(91)90230-w. 202–207. free.
• Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.

Notes and References

1. J. Bourgain, H ^{infty} is a Grothendieck space, Studia Math., 75 (1983), 193–216.