Banach lattice
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In mathematics, specifically in functional analysis and order theory, a Banach lattice is a normed lattice with a norm such that is a Banach space and for all the implication implies holds, where as usual
Examples and constructions[]
- together with its absolute value as a norm, is a Banach lattice.
- Let be a topological space, a Banach lattice and the space of bounded, continuous functions from to with norm becomes a Banach lattice with the pointwise order if and only if for every
Properties[]
The continuous dual space of a Banach lattice is equal to its order dual.[1]
See also[]
- Banach space – Normed vector space that is complete
- Normed vector lattice
- Riesz space – Partially ordered vector space, ordered as a lattice
- Lattice (order) – Set whose pairs have minima and maxima
References[]
- ^ Schaefer & Wolff 1999, pp. 234–242.
- Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. Vol. 50. American Mathematical Society. ISBN 0-8218-2146-6.
Bibliography[]
- 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.
Categories:
- Functional analysis
- Order theory
- Mathematical analysis stubs