Banach 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 [math]\displaystyle{ {|x|\leq|y|}\Rightarrow{\|x\|\leq\|y\|} }[/math] holds, where the absolute value |·| is defined as [math]\displaystyle{ |x| = x \vee -x := \sup\{x, -x\}\text{.} }[/math]
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:
- Template:Mathbb, 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 [math]\displaystyle{ \|f\|_{\infty} = \sup_{x \in X} \|f(x)\|_Y\text{.} }[/math] Then 𝒞(X,Y) is a Banach lattice under the pointwise partial order: [math]\displaystyle{ {f \leq g}\Leftrightarrow(\forall x\in X)(f(x)\leq g(x))\text{.} }[/math]
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.[3]
Every Banach lattice admits a continuous approximation to the identity.[4]
Abstract (L)-spaces
A Banach lattice satisfying the additional condition [math]\displaystyle{ {f,g\geq0}\Rightarrow\|f+g\|=\|f\|+\|g\| }[/math] 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
- 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
- ↑ Birkhoff 1948, p. 246.
- ↑ Kania, Tomasz (12 April 2017). Answer to "Banach space that is not a Banach lattice" (accessed 13 August 2022). Mathematics StackExchange. StackOverflow.
- ↑ Schaefer & Wolff 1999, pp. 234–242.
- ↑ Birkhoff 1948, p. 251.
- ↑ Birkhoff 1948, pp. 250,254.
- ↑ Birkhoff 1948, pp. 269-271.
Bibliography
- Abramovich, Yuri A.; Aliprantis, C. D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. 50. American Mathematical Society. ISBN 0-8218-2146-6.
- Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. https://hdl.handle.net/2027/iau.31858027322886.
- 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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