Chapter

Analysis in Banach Spaces

Volume 63 of the series Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics pp 1-66

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Bochner spaces

  • Tuomas HytönenAffiliated withDepartment of Mathematics and Statistics, University of Helsinki Email author 
  • , Jan van NeervenAffiliated withDelft Institute of Applied Mathematics, Delft University of Technology
  • , Mark VeraarAffiliated withDelft Institute of Applied Mathematics, Delft University of Technology
  • , Lutz WeisAffiliated withDepartment of Mathematics, Karlsruhe Institute of Technology

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Abstract

This chapter sets up the general framework in which we work throughout these volumes. After introducing the relevant notions of measurability for functions taking values in a Banach space, we proceed to define the Bochner integral and the Bochner spaces L p (S;X), which are the vector-valued counterparts of the Lebesgue integral and the classical L p -spaces, respectively. We also briefly discuss the weaker Pettis integral. The chapter concludes with a detailed investigation of duality of the Bochner spaces and the related Radon–Nikodým property.