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Normed vector lattice

From Wikipedia, the free encyclopedia

In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set.[1] Normed lattices are important in the theory of topological vector lattices. They are closely related to Banach vector lattices, which are normed vector lattices that are also Banach spaces.

Properties

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Every normed lattice is a locally convex vector lattice.[1]

The strong dual of a normed lattice is a Banach lattice with respect to the dual norm and canonical order. If it is also a Banach space then its continuous dual space is equal to its order dual.[1]

Examples

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Every Banach lattice is a normed lattice.

See also

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References

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  1. ^ a b c Schaefer & Wolff 1999, pp. 234–242.

Bibliography

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  • 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.