Suppose is a real (or complex) vector space with a subset .
If for any real , then is called a cone.
If the origin belongs to a cone, then the cone is said to be pointed. Otherwise, the cone is blunt.
A pointed cone is salient, if it contains no -dimensional vector subspace of .
If is a cone for some in , then is a cone with vertex at .
A convex pointed cone is called a wedge.
A proper cone is a convex cone with vertex at , such that . A slightly more specific definition of a proper cone is this entry (http://planetmath.org/ProperCone), but it requires the vector space to be topological.
A cone is said to be generating if . In this case, is said to be generated by .
A cone is convex iff .
A cone containing is a cone with vertex at . As a result, a wedge is a cone with vertex at .
The only cones that are subspaces at the same time are wedges.
- 1 M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
- 2 J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company, 1966.
- 3 R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.
- 4 I.M. Glazman, Ju.I. Ljubic, Finite-Dimensional Linear Analysis, A systematic Presentation in Problem Form, Dover Publications, 2006.
|Date of creation||2013-03-22 15:32:58|
|Last modified on||2013-03-22 15:32:58|
|Last modified by||matte (1858)|
|Defines||cone with vertex|