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The property of a [[Banach space|Banach space]]  $  X $
that every continuous operator  $  T : X \rightarrow Y $
sending bounded sets of  $  X $
into relatively weakly compact sets of  $  Y $(
called weakly compact operators) also transforms weakly compact sets of  $  X $
into norm-compact sets of  $  Y $(
such operators are called completely continuous; cf. also [[Completely-continuous operator|Completely-continuous operator]]). In short, it requires that weakly compact operators on  $  X $
are completely continuous.

Equivalently, given weakly convergent sequences  $  ( x _ {n} ) $
in  $  X $
and  $  ( f _ {n} ) $
in its topological dual  $  X  ^ {*} $, 
the sequence  $  ( f _ {n} ( x _ {n} ) ) _ {n} $
also converges. Contrary to intuition this does not always happen. For example, if  $  ( e _ {n} ) $
denotes the canonical basis of  $  l _ {2} $, 
then  $  ( e _ {n} ) $
is weakly convergent to zero although  $  e _ {n} ( e _ {n} ) = 1 $.

The property was isolated and defined by A. Grothendieck [[#References|[a7]]] after the following classical result of N. Dunford and B.J. Pettis [[#References|[a5]]]: For any measure  $  \mu $
and any Banach space  $  Y $, 
every weakly compact operator  $  L _ {1} ( \mu ) $
into  $  Y $
is completely continuous.

This result has its roots in examples of Sirvint, S. Kakutani, Y. Mimura and K. Yosida concerning weakly compact non-compact operators on  $  L _ {1} ( 0,1 ) $
which could be proven to have a compact square. The main examples of spaces having the Dunford–Pettis property are the spaces  $  C ( K ) $
of continuous functions on a compact space and the spaces  $  L _ {1} ( \mu ) $
of integrable functions on a measure space, as well as complemented subspaces of these spaces. Other classical function spaces having the Dunford–Pettis property are: the Hardy space  $  H  ^  \infty  $
and its higher duals (cf. also [[Hardy spaces|Hardy spaces]]); the quotient space  $  L _ {1} /H  ^ {1} $
and its higher duals (the space  $  H  ^ {1} $
itself does not have the Dunford–Pettis property, nor does its dual BMO or its pre-dual VMO) (cf. also [[BMO-space| $  { \mathop{\rm BMO} } $-
space]]; [[VMO-space| $  { \mathop{\rm VMO} } $-
space]]); the ball algebra, the poly-disc algebra and their duals, and the spaces  $  C  ^ {k} ( T  ^ {n} ) $
of  $  k $-
smooth functions on the  $  n $-
dimensional torus.

A classical survey on the topic is [[#References|[a4]]]. Many of the open problems stated there have been solved by now, mainly by J. Bourgain [[#References|[a2]]], [[#References|[a3]]], who introduced new techniques for working with the Dunford–Pettis property, and by M. Talagrand [[#References|[a8]]], who gave an example of a space  $  X $
with the Dunford–Pettis property such that  $  C ( K,X ) $
and  $  L _ {1} ( \mu,X  ^ {*} ) $
fail the Dunford–Pettis property.

The Dunford–Pettis property is not easy to work with, nor is it well understood. In general, it is difficult to prove that a given concrete space has the property; quoting J. Diestel:  "I know of no case where the reward (when it comes) is easily attained" . On the question of structure theorems, many open problems remain. One of the most striking is as follows. When does the dual of a space that has the Dunford–Pettis property have the Dunford–Pettis property? It is clear that if  $  X  ^ {*} $
has the Dunford–Pettis property, then so does  $  X $. 
From Rosenthal's  $  l _ {1} $
theorem it follows that if  $  X $
has the Dunford–Pettis property and does not contain  $  l _ {1} $, 
then  $  X  ^ {*} $
has the Dunford–Pettis property. Stegall has shown that although the space  $  l _ {1} ( l _ {2}  ^ {n} ) $
has the Dunford–Pettis property (since weakly convergent sequences are norm convergent), its dual  $  l _  \infty  ( l _ {2}  ^ {n} ) $
does not have the Dunford–Pettis property (because it contains complemented copies of  $  l _ {2} $).

A [[Reflexive space|reflexive space]] does not have the Dunford–Pettis property unless it is finite-dimensional. The Grothendieck spaces  $  C ( \Omega ) $, 
$  L  ^  \infty  ( \mu ) $, 
$  B ( S, \Sigma ) $, 
and  $  H  ^  \infty  ( D ) $(
cf. [[Grothendieck space|Grothendieck space]]) also possess the Dunford–Pettis property (see [[#References|[a9]]], [[#References|[a10]]]).

A Banach space  $  X $
is a Grothendieck space with the Dunford–Pettis property if and only if every weak- $  * $
convergent sequence in  $  X  ^ {*} $
converges weakly and uniformly on weakly compact subsets of  $  X $, 
if and only if every bounded [[Linear operator|linear operator]] from  $  X $
into  $  c _ {0} $
is weakly compact and maps weakly compact sets into norm-compact sets.

An interesting phenomenon about Grothendieck spaces with the Dunford–Pettis property is that in many cases strong convergence of operators on such a space (cf. also [[Strong topology|Strong topology]]) implies [[Uniform convergence|uniform convergence]]. For example, let  $  X $
be a Grothendieck space with the Dunford–Pettis property. Then:

1)  $  X $
does not have a Schauder decomposition, or equivalently, if a sequence of projections  $  \{ P _ {n} \} $
on  $  X $
converges weakly to the identity operator  $  I $, 
then  $  P _ {n} = I $
for  $  n $
sufficiently large;

2) if the Cesáro mean  $  n ^ {- 1 } \sum _ {k = 0 }  ^ {n - 1 } T  ^ {k} $
of an operator  $  T $
on  $  X $
converges strongly, then it converges uniformly;

3) all  $  C _ {0} $-
semi-groups on  $  X $
are norm-continuous (see [[#References|[a9]]], [[#References|[a10]]]);

4) all strongly continuous cosine operator functions on  $  X $
are norm-continuous [[#References|[a11]]];

5) for general ergodic systems on  $  X $, 
in particular,  $  C _ {0} $-
semi-groups and cosine operator functions, strong ergodicity implies uniform ergodicity (see [[#References|[a12]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bourgain,   "On the Dunford–Pettis property"  ''Proc. Amer. Math. Soc.'' , '''81'''  (1981)  pp. 265–272</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bourgain,   "New Banach space properties of the disc algebra and $G^\infty$"  ''Acta Math.'' , '''152'''  (1984)  pp. 1–48</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Bourgain,   "The Dunford–Pettis property for the ball-algebras, the polydisc algebra, and the Sobolev spaces"  ''Studia Math.'' , '''77'''  (1984)  pp. 245–253</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Diestel,   "A survey or results related to the Dunford–Pettis property" , ''Contemp. Math.'' , '''2''' , Amer. Math. Soc.  (1980)  pp. 15–60</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Dunford,   B.J. Pettis,   "Linear operations on summable functions"  ''Trans. Amer. Math. Soc.'' , '''47'''  (1940)  pp. 323–392</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''I. General theory''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Grothendieck,   "Sur les applications linéaires faiblement compactes d'espaces de type $C(K)$"  ''Canad. J. Math.'' , '''5'''  (1953)  pp. 129–173</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Talagrand,   "La propriété de Dunford–Pettis dans $C(K,E)$ et $L_1(E)"  ''Israel J. Math.'' , '''44'''  (1983)  pp. 317–321</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  H.P. Lotz,   "Tauberian theorems for operators on $L^\infty$ and similar spaces" , ''Functional Analysis III. Surveys and Recent Results'' , North-Holland  (1984)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H.P. Lotz,   "Uniform convergence of operators on $L^\infty$ and similar spaces"  ''Math. Z.'' , '''190'''  (1985)  pp. 207–220</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  S.-Y. Shaw,   "Asymptotic behavior of pseudoresolvents on some Grothendieck spaces"  ''Publ. RIMS Kyoto Univ.'' , '''24'''  (1988)  pp. 277–282</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S.-Y. Shaw,   "Uniform convergence of ergodic limits and approximate solutions"  ''Proc. Amer. Math. Soc.'' , '''114'''  (1992)  pp. 405–411</TD></TR></table>