Dieudonné's theorem

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In mathematics, Dieudonné's theorem, named after Jean Dieudonné, is a theorem on when the Minkowski sum of closed sets is closed.

Statement of theorem[edit]

Let nonempty closed convex sets A,B \subset X a locally convex space, if either A or B is locally compact and \operatorname{recc}(A) \cap \operatorname{recc}(B) (where \operatorname{recc} gives the recession cone) is a linear subspace, then A - B is closed.[1][2]

References[edit]

  1. ^ J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann. 163. 
  2. ^ Zălinescu, Constantin (2002). Convex analysis in general vector spaces (J). River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556.