Banach lattice

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In mathematics, specifically in functional analysis and order theory, a Banach lattice ${\displaystyle (X,\|\cdot \|)}$ is a normed lattice with a norm ${\displaystyle \|\cdot \|}$ such that ${\displaystyle (X,\|\cdot \|)}$ is a Banach space and for all ${\displaystyle x,y\in X}$ the implication ${\displaystyle |x|\leq |y|}$ implies ${\displaystyle \|x\|\leq \|y\|}$ holds, where as usual ${\displaystyle |x|:=x\vee -x.}$

Examples and constructions[]

• ${\displaystyle \mathbb {R} ,}$ together with its absolute value as a norm, is a Banach lattice.
• Let ${\displaystyle X}$ be a topological space, ${\displaystyle Y}$ a Banach lattice and ${\displaystyle {\mathcal {C}}(X,Y)}$ the space of bounded, continuous functions from ${\displaystyle X}$ to ${\displaystyle Y}$ with norm ${\displaystyle \|f\|_{\infty }:=\sup _{x\in X}\|f(x)\|_{Y}.}$ ${\displaystyle {\mathcal {C}}(X,Y)}$ becomes a Banach lattice with the pointwise order ${\displaystyle f\leq g}$ if and only if ${\displaystyle f(x)\leq g(x)}$ for every ${\displaystyle x\in X.}$

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[]

• 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.

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