Background Material

  • J. Frédéric Bonnans
  • Alexander Shapiro
Part of the Springer Series in Operations Research book series (ORFE)


This chapter recalls some basic results from topology and functional analysis, as well as tools that play an essential role in the perturbation theory of convex and nonconvex optimization problems. We present some of the results in a fairly general framework of locally convex topological vector spaces, although all optimization problems we deal with use the Banach space framework. The reason for this is that a Banach space X (endowed with the strong topology) is the dual of the dual space X* (endowed with the weak* topology). In the locally convex space setting, we have a complete symmetry between primal and dual spaces. Therefore, duality results are obtained in both directions. For some results we give proofs, while other (classical) results are only stated. Their proofs can be found in almost any standard text on functional analysis.


Banach Space Dual Problem Lower Semicontinuous Topological Vector Space Tangent Cone 
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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • J. Frédéric Bonnans
    • 1
  • Alexander Shapiro
    • 2
  1. 1.INRIA-RocquencourtDomaine de VoluceauLe Chesnay CedexFrance
  2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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