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
Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a vector lattice."[1] In particular:
Examples of non-lattice Banach spaces are now known; James' space is one such.[2]
The continuous dual space of a Banach lattice is equal to its order dual.[3]
Every Banach lattice admits a continuous approximation to the identity.[4]
A Banach lattice satisfying the additional condition