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, yX, 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."[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]

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]).[5] The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.[6]

See also


  1. ^ Birkhoff 1948, p. 246.
  2. ^ Kania, Tomasz (12 April 2017). Answer to "Banach space that is not a Banach lattice" (accessed 13 August 2022). Mathematics StackExchange. StackOverflow.
  3. ^ Schaefer & Wolff 1999, pp. 234–242.
  4. ^ Birkhoff 1948, p. 251.
  5. ^ Birkhoff 1948, pp. 250, 254.
  6. ^ Birkhoff 1948, pp. 269–271.