In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Dieudonné's theorem, named after
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous ...
, is a
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
on when the
Minkowski sum
In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'':
A + B = \
The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...
of
closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s is closed.
Statement
Let
be a
locally convex space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
and
nonempty closed
convex set
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
s. If either
or
is
locally compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
and
(where
gives the
recession cone
In mathematics, especially convex analysis, the recession cone of a set A is a cone containing all vectors such that A ''recedes'' in that direction. That is, the set extends outward in all the directions given by the recession cone.
Mathematica ...
) is a
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
, then
is closed.
References
Convex analysis
Theorems in functional analysis
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