Special case

From formulasearchengine
Jump to navigation Jump to search

In mathematics, specifically functional analysis, the barrier cone is a cone associated to any non-empty subset of a Banach space. It is closely related to the notions of support functions and polar sets.

Definition

Let X be a Banach space and let K be a non-empty subset of X. The barrier cone of K is the subset b(K) of X, the continuous dual space of X, defined by

b(K):={X|supxK,x<+}.

Related notions

The function

σK:supxK,x,

defined for each continuous linear functional on X, is known as the support function of the set K; thus, the barrier cone of K is precisely the set of continuous linear functionals for which σK() is finite.

The set of continuous linear functionals for which σK() ≤ 1 is known as the polar set of K. The set of continuous linear functionals for which σK() ≤ 0 is known as the (negative) polar cone of K. Clearly, both the polar set and the negative polar cone are subsets of the barrier cone.

References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534