Abstract

We establish a Fenchel-Moreau type theorem for proper convex functions f ⁣:X→Lˉ0f\colon X\to \bar{L}^0, where (X,Y,⟨⋅,⋅⟩)(X, Y, \langle \cdot,\cdot \rangle) is a dual pair of Banach spaces and Lˉ0\bar L^0 is the space of all extended real-valued functions on a σ\sigma-finite measure space. We introduce the concept of stable lower semi-continuity which is shown to be equivalent to the existence of a dual representation f(x)=sup⁡y∈L0(Y){⟨x,y⟩−f∗(y)},x∈X,\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y \rangle - f^\ast(y)\right\}, \quad x\in X,} where L0(Y)L^0(Y) is the space of all strongly measurable functions with values in YY, and ⟨⋅,⋅⟩\langle \cdot,\cdot \rangle is understood pointwise almost everywhere. The proof is based on a conditional extension result and conditional functional analysis.

Author information

Contact details are reproduced from the original publication and may be historical.

Samuel Drapeau

Shanghai Jiao Tong University, School of Mathematical Sciences, and: China Academy of Financial Research, 211 West Huaihai Road, Shanghai, China

sdrapeau@saif.sjtu.edu.cn

Michael Kupper

Dept. of Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany

kupper@uni-konstanz.de

Keywords

  • Fenchel-Moreau theorem
  • vector duality
  • semi-continuous extension
  • conditional functional analysis

Mathematics Subject Classification

46A20, 03C90, 46B22

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Suggested citation

S. Drapeau, A. Jamneshan, M. Kupper. “A Fenchel-Moreau Theorem for L̄^0-Valued Functions.” Journal of Convex Analysis 26 (2019), No. 2, 593–603. https://doi.org/10.68381/jca26031

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.