In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space whose positive cone is a closed subset of X.[1] Ordered TVSes have important applications in spectral theory.

Normal cone

Main article: Normal cone (functional analysis)

If C is a cone in a TVS X then C is normal if , where is the neighborhood filter at the origin, , and is the C-saturated hull of a subset U of X.[2]

If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:[2]

  1. C is a normal cone.
  2. For every filter in X, if then .
  3. There exists a neighborhood base in X such that implies .

and if X is a vector space over the reals then also:[2]

  1. There exists a neighborhood base at the origin consisting of convex, balanced, C-saturated sets.
  2. There exists a generating family of semi-norms on X such that for all and .

If the topology on X is locally convex then the closure of a normal cone is a normal cone.[2]

Properties

If C is a normal cone in X and B is a bounded subset of X then is bounded; in particular, every interval is bounded.[2] If X is Hausdorff then every normal cone in X is a proper cone.[2]

Properties

  1. the order of X is regular.
  2. C is sequentially closed for some Hausdorff locally convex TVS topology on X and distinguishes points in X
  3. the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.

See also

References

  1. ^ a b c Schaefer & Wolff 1999, pp. 222–225.
  2. ^ a b c d e f Schaefer & Wolff 1999, pp. 215–222.