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,
Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a vector lattice.
[3] Every Banach lattice admits a continuous approximation to the identity.
[4] A Banach lattice satisfying the additional condition
Such spaces, under the assumption of separability, are isomorphic to closed sublattices of L1([0,1]).
[5] The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.