Bipolar theorem

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (i.e. the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.[1]:76–77

Preliminaries[edit]

Suppose that X is a topological vector space (TVS) with a continuous dual space and let for all xX and . The convex hull of a set A, denoted by co(A), is the smallest convex set containing A. The convex balanced hull of a set A is the smallest convex balanced set containing A.

The polar of a subset A of X is defined to be:

while the prepolar of a subset B of is:

.

The bipolar of a subset A of X, often denoted by A∘∘ is the set

.

Statement in functional analysis[edit]

Let denote the weak topology on X (i.e. the weakest TVS topology on X making all linear functionals in continuous).

The bipolar theorem:[2] The bipolar of a subset A of X is equal to the -closure of the convex balanced hull of A.

Statement in convex analysis[edit]

The bipolar theorem:[1]:54[3] For any nonempty cone A in some linear space X, the bipolar set A∘∘ is given by:
.

Special case[edit]

A subset C of X is a nonempty closed convex cone if and only if C++ = C∘∘ = C when C++ = (C+)+, where A+ denotes the positive dual cone of a set A.[3][4] Or more generally, if C is a nonempty convex cone then the bipolar cone is given by

C∘∘ = cl(C).

Relation to the Fenchel–Moreau theorem[edit]

Let

be the indicator function for a cone C. Then the convex conjugate,

is the support function for C, and . Therefore, C = C∘∘ if and only if f = f**.[1]:54[4]

See also[edit]

References[edit]

  1. ^ a b c Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701. CS1 maint: discouraged parameter (link)
  2. ^ Narici & Beckenstein 2011, pp. 225-273.
  3. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011. CS1 maint: discouraged parameter (link)
  4. ^ a b Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866. CS1 maint: discouraged parameter (link)

Bibliography[edit]