Topological vector lattice

Summary

In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) that has a partial order making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.[1] Ordered vector lattices have important applications in spectral theory.

Definition edit

If   is a vector lattice then by the vector lattice operations we mean the following maps:

  1. the three maps   to itself defined by  ,  ,  , and
  2. the two maps from   into   defined by   and .

If   is a TVS over the reals and a vector lattice, then   is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.[1]

If   is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous.[1]

If   is a topological vector space (TVS) and an ordered vector space then   is called locally solid if   possesses a neighborhood base at the origin consisting of solid sets.[1] A topological vector lattice is a Hausdorff TVS   that has a partial order   making it into vector lattice that is locally solid.[1]

Properties edit

Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.[1] Let   denote the set of all bounded subsets of a topological vector lattice with positive cone   and for any subset  , let   be the  -saturated hull of  . Then the topological vector lattice's positive cone   is a strict  -cone,[1] where   is a strict  -cone means that   is a fundamental subfamily of   that is, every   is contained as a subset of some element of  ).[2]

If a topological vector lattice   is order complete then every band is closed in  .[1]

Examples edit

The Lᵖ spaces ( ) are Banach lattices under their canonical orderings. These spaces are order complete for  .

See also edit

References edit

  1. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 234–242.
  2. ^ Schaefer & Wolff 1999, pp. 215–222.

Bibliography edit

  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.