Banach space with a compatible structure of a lattice
In the mathematical disciplines of in functional analysis and order theory, a Banach lattice (X,‖·‖) is a complete normed vector space with a lattice order, such that for all x, y ∈ X, 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:
- ℝ, together with its absolute value as a norm, is a Banach lattice.
- Let X be a topological space, Y a Banach lattice and 𝒞(X,Y) the space of continuous bounded functions from X to Y with norm
Then 𝒞(X,Y) is a Banach lattice under the pointwise partial order:
Examples of non-lattice Banach spaces are now known; James' space is one such.[2]
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 L1([0,1]). The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.