Dieudonné's theorem

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

Statement

Let $X$ be a locally convex space and $A,B\subset X$ nonempty closed convex sets. 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]

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References

J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.
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.

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[1] J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.

[Zalinescu-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.

Functional analysis (topics, glossary)
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Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory