Banach lattice explained

In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order,

\leq , such that for all, the implication $$$$

\leq

y

\Rightarrow holds, where the absolute value is defined as $$|x|\; =\; x\; \backslash vee\; -x\; :=\; \backslash sup\backslash \backslash text$$

Examples and constructions

Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a vector lattice." In particular:

• , together with its absolute value as a norm, is a Banach lattice.
• Let be a topological space, a Banach lattice and the space of continuous bounded functions from to with norm $$\backslash |f\backslash |\_\; =\; \backslash sup\_\; \backslash |f(x)\backslash |\_Y\backslash text$$ Then is a Banach lattice under the pointwise partial order: $$\backslash Leftrightarrow(\backslash forall\; x\backslash in\; X)(f(x)\backslash leq\; g(x))\backslash text$$

Examples of non-lattice Banach spaces are now known; James' space is one such.^[1]

Properties

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

Every Banach lattice admits a continuous approximation to the identity.

Abstract (L)-spaces

A Banach lattice satisfying the additional condition $$\backslash Rightarrow\backslash |f+g\backslash |=\backslash |f\backslash |+\backslash |g\backslash |$$ is called an abstract (L)-space. Such spaces, under the assumption of separability, are isomorphic to closed sublattices of . The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.

Bibliography

• Book: Abramovich, Yuri A.. Aliprantis, C. D.. 2002. An Invitation to Operator Theory. Graduate Studies in Mathematics. 50. American Mathematical Society. 0-8218-2146-6.
• Book: Birkhoff, Garrett. AMS Colloquium Publications 25. Lattice Theory. Garrett Birkhoff. Revised. AMS. New York City. 1948. 2027/iau.31858027322886 . HathiTrust.

Notes and References

1. Kania, Tomasz (12 April 2017). Answer to "Banach space that is not a Banach lattice" (accessed 13 August 2022). Mathematics StackExchange. StackOverflow.