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 [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 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 (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.

See also

Footnotes

Bibliography