Dunford–Pettis property

A Banach space ${\displaystyle X}$ has the Dunford–Pettis property if every continuous weakly compact operator ${\displaystyle T:X\to Y}$ from ${\displaystyle X}$ into another Banach space ${\displaystyle Y}$ transforms weakly compact sets in ${\displaystyle X}$ into norm-compact sets in ${\displaystyle Y}$ (such operators are called completely continuous). An important equivalent definition is that for any weakly convergent sequences ${\displaystyle x_{1},x_{2},\ldots }$ of ${\displaystyle X}$ and ${\displaystyle f_{1},f_{2},\ldots }$ of the dual space ${\displaystyle X^{*},}$ converging (weakly) to ${\displaystyle x}$ and ${\displaystyle f,}$ the sequence ${\displaystyle f_{1}(x_{1}),f_{2}(x_{2}),\ldots ,f_{n}(x_{n}),\ldots }$ converges to ${\displaystyle f(x).}$

• The second definition may appear counterintuitive at first, but consider an orthonormal basis ${\displaystyle e_{n}}$ of an infinite-dimensional, separable Hilbert space ${\displaystyle H.}$ Then ${\displaystyle e_{n}\to 0}$ weakly, but for all ${\displaystyle n}$
  $${\displaystyle \langle e_{n},e_{n}\rangle =1.}$$

  
  Thus separable infinite-dimensional Hilbert spaces cannot have the Dunford–Pettis property.
• Consider as another example the space ${\displaystyle L^{p}(-\pi ,\pi )}$ where ${\displaystyle 1<p<\infty .}$ The sequences ${\displaystyle x_{n}=e^{inx}}$ in ${\displaystyle L^{p}}$ and ${\displaystyle f_{n}=e^{inx}}$ in ${\displaystyle L^{q}=\left(L^{p}\right)^{*}}$ both converge weakly to zero. But
  $${\displaystyle \langle f_{n},x_{n}\rangle =\int \limits _{-\pi }^{\pi }1\,{\rm {d}}x=2\pi .}$$

  
• More generally, no infinite-dimensional reflexive Banach space may have the Dunford–Pettis property. In particular, an infinite-dimensional Hilbert space and more generally, Lp spaces with ${\displaystyle 1<p<\infty }$ do not possess this property.

• If ${\displaystyle K}$ is a compact Hausdorff space, then the Banach space ${\displaystyle C(K)}$ of continuous functions with the uniform norm has the Dunford–Pettis property.

• Grothendieck space

• Bourgain, Jean (1981), "On the Dunford–Pettis property", Proceedings of the American Mathematical Society, 81 (2): 265–272, doi:10.2307/2044207, JSTOR 2044207
• Grothendieck, Alexander (1953), "Sur les applications linéaires faiblement compactes d'espaces du type C(K)", Canadian Journal of Mathematics, 5: 129–173, doi:10.4153/CJM-1953-017-4
• JMF Castillo, SY Shaw (2001) [1994], "Dunford–Pettis property", Encyclopedia of Mathematics, EMS Press
• Lin, Pei-Kee (2004), Köthe-Bochner Function Spaces, Birkhäuser, ISBN 0-8176-3521-1, OCLC 226084233
• Randrianantoanina, Narcisse (1997), "Some remarks on the Dunford-Pettis property" (PDF), Rocky Mountain Journal of Mathematics, 27 (4): 1199–1213, doi:10.1216/rmjm/1181071869, S2CID 15539667