Banach lattice

In the mathematical disciplines of in functional analysis and order theory, a Banach lattice is a complete normed vector space with a lattice order, such that for all , the implication $\Rightarrow$ holds, where the absolute value is defined as $, x, = x \vee -x\text$

Examples and constructions

Banach lattices are extremely common in functional analysis, and "every known example n 1948of a Banach space asalso 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 $\, f\, _ = \sup_ \, f(x)\, _Y\text$ Then is a Banach lattice under the pointwise partial order: $\Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text$

Examples of non-lattice Banach spaces are now known; James' space is one such.Kania, Tomasz (12 April 2017).
Answer
to "Banach space that is not a Banach lattice" (accessed 13 August 2022). ''Mathematics StackExchange''. StackOverflow.

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 $\Rightarrow\, f+g\, =\, f\, +\, g\,$ is called an abstract (L)-space. Such spaces are necessarily uniformly convex, and separable ones are isomorphic to closed sublattices of . The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.

See also

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Footnotes

Bibliography

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Functional analysis
Order theory