Grothendieck space

In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space X such that for all separable Banach spaces 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.

Every reflexive Banach space is a Grothendieck space. Conversely, 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).

See also

• Dunford–Pettis property

References

• Shaw, S.-Y. (2001), "G/g110250", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4