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User:Zfeinst/Closure of Sum of Closed Sets

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If are closed sets in a topological vector space then is closed if

  1. Either or is compact set
  2. Dieudonne's theorem: Let nonempty closed convex sets a locally convex space, if either or is locally compact and (where gives the recession cone) is a linear subspace, then is closed.[1][2]
  3. Let nonempty closed convex sets such that for any then , then is closed.[3][4]
  4. Let nonempty closed convex sets a reflexive Banach space contain no lines, if then is closed (where ). In fact, if this condition is satisfied then any two closed convex uniform perturbations and fulfilling and then is closed.[5]
  5. ...

References

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  1. ^ J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.
  2. ^ Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556.
  3. ^ Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6.
  4. ^ Kim C. Border. "Sums of sets, etc" (pdf). Retrieved March 7, 2012.
  5. ^ Adly, Samir; Ernst, Emil; Théra, Michel (2003). "On the closedness of the algrebraic difference of closed convex sets". J. Math. Pures Appl.. 82 (9): 1219–1249.