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
![{\displaystyle {|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|))](https://wikimedia.org/api/rest_v1/media/math/render/svg/6917b88005f16d66d4174b94f4ab03d0cce66488)
holds, where the absolute value |·| is defined as ![{\displaystyle |x|=x\vee -x:=\sup\{x,-x\}{\text{.))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5979ac5fdc16260f11eafde48cfcd93a18ab9d5f)
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
![{\displaystyle \|f\|_{\infty }=\sup _{x\in X}\|f(x)\|_{Y}{\text{.))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0d98e144e94071ebfe56cd22e9e5c21efe58c)
Then 𝒞(X,Y) is a Banach lattice under the pointwise partial order: ![{\displaystyle {f\leq g}\Leftrightarrow (\forall x\in X)(f(x)\leq g(x)){\text{.))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb21aabc12fc20b31383fcb8e7fac7ee054a8999)
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
![{\displaystyle {f,g\geq 0}\Rightarrow \|f+g\|=\|f\|+\|g\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3616da14067722da1232fb45b6d5a6a86860341)
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.