Banach lattice

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 ${\displaystyle |x|\leq |y|\Rightarrow \|x\|\leq \|y\|}$ 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 ${\displaystyle f\leq g:\Leftrightarrow \forall x\in X:f(x)\leq g(x)}$.

Properties

The continuous dual space of a Banach lattice is equal to its order dual.[1]

See also

• Banach space
• Normed vector lattice
• Riesz space
• Lattice (order)

References

1. Schaefer & Wolff 1999, pp. 234–242.

• Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. 50. American Mathematical Society. ISBN 0-8218-2146-6.
• 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.