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, ≤ {\displaystyle \leq } , such that for all x, y ∈ X, the implication | x | ≤ | y | ⇒ ‖ x ‖ ≤ ‖ y ‖ {\displaystyle {|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}} holds, where the absolute value |·| is defined as | x | = x ∨ − x := sup { x , − x } . {\displaystyle |x|=x\vee -x:=\sup\{x,-x\}{\text{.}}}
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 ‖ f ‖ ∞ = sup x ∈ X ‖ f ( x ) ‖ Y . {\displaystyle \|f\|_{\infty }=\sup _{x\in X}\|f(x)\|_{Y}{\text{.}}} Then 𝒞(X,Y) is a Banach lattice under the pointwise partial order: f ≤ g ⇔ ( ∀ x ∈ X ) ( f ( x ) ≤ g ( x ) ) . {\displaystyle {f\leq g}\Leftrightarrow (\forall x\in X)(f(x)\leq g(x)){\text{.}}}
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 f , g ≥ 0 ⇒ ‖ f + g ‖ = ‖ f ‖ + ‖ g ‖ {\displaystyle {f,g\geq 0}\Rightarrow \|f+g\|=\|f\|+\|g\|} 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.
See also Banach space – Normed vector space that is complete Normed vector lattice Riesz space – Partially ordered vector space, ordered as a lattice Lattice (order) – Set whose pairs have minima and maxima
Footnotes
Bibliography Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. Vol. 50. American Mathematical Society. ISBN 0-8218-2146-6. Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.