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 holds, where the absolute value is defined as

Examples and constructions

Banach lattices are extremely common in functional analysis, and "every known example of a Banach space 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 Then is a Banach lattice under the pointwise partial order:

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

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 is called an abstract -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 -spaces.