View source for Grothendieck space

In [[mathematics]], a '''Grothendieck space''', named after [[Alexander Grothendieck]], is a [[Banach space]] <math>X</math> in which every sequence in its [[continuous dual space]] <math>X^{\prime}</math> that converges in the [[weak-* topology]] <math>\sigma\left(X^{\prime}, X\right)</math>  (also known as the [[topology of pointwise convergence]]) will also converge when <math>X^{\prime}</math> is endowed with <math>\sigma\left(X^{\prime}, X^{\prime \prime}\right),</math> which is the [[weak topology]] induced on <math>X^{\prime}</math> 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 <math>X</math> be a Banach space. Then the following conditions are equivalent:
# <math>X</math> is a Grothendieck space,
# for every [[separable space|separable]] Banach space <math>Y,</math> every [[Bounded operator|bounded linear operator]] from <math>X</math> to <math>Y</math> is [[weak topology|weakly compact]], that is, the image of a bounded subset of <math>X</math> is a weakly compact subset of <math>Y.</math>
# for every weakly compactly generated Banach space <math>Y,</math>  every bounded linear operator from <math>X</math> to <math>Y</math> is [[Weak topology|weakly compact]].
# every weak*-continuous function on the dual <math>X^{\prime}</math> is weakly Riemann integrable.

== Examples ==

* Every [[Reflexive space|reflexive]] Banach space is a Grothendieck space. Conversely, it is a consequence of the [[Eberlein–Šmulian theorem]] that a separable Grothendieck space <math>X</math> must be reflexive, since the identity from <math>X \to X</math> is weakly compact in this case.
* Grothendieck spaces which are not reflexive include the space <math>C(K)</math> of all continuous functions on a [[Extremally disconnected space|Stonean]] compact space <math>K,</math> and the space <math>L^{\infty}(\mu)</math> for a [[positive measure]] <math>\mu</math> (a Stonean compact space is a [[Hausdorff space|Hausdorff]] compact space in which the [[Closure (mathematics)|closure]] of every [[open set]] is open).
* [[Jean Bourgain]] proved that the space <math>H^{\infty}</math> of bounded holomorphic functions on the disk is a Grothendieck space.<ref>J. Bourgain, <math>H^{\infty}</math> is a Grothendieck space, ''Studia Math.'', '''75''' (1983), 193–216.</ref>

== See also ==

* {{annotated link|Dunford–Pettis property}}

==References==

{{reflist}}

* J. Diestel, ''Geometry of Banach spaces'', Selected Topics, Springer, 1975.
* J. Diestel, J. J. Uhl: ''Vector measures''. Providence, R.I.: American Mathematical Society, 1977. {{ISBN|978-0-8218-1515-1}}.
*{{eom|title=Grothendieck space|first=S.-Y. |last=Shaw}}
* {{cite journal|last=Khurana|first=Surjit Singh|title=Grothendieck spaces, II|journal=Journal of Mathematical Analysis and Applications|publisher=Elsevier BV|volume=159|issue=1|year=1991|issn=0022-247X|doi=10.1016/0022-247x(91)90230-w|pages=202–207|doi-access=free}} <!--{{sfn|Khurana|1991|pp=202–207}}-->
* Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics,  2017.

{{Banach spaces}}
{{Functional analysis}}
{{TopologicalVectorSpaces}}

{{mathanalysis-stub}}

[[Category:Banach spaces]]