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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:

• 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.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 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]).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

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.

References

1. Birkhoff 1948, p. 246. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. https://hdl.handle.net/2027/iau.31858027322886 ↩

2. Kania, Tomasz (12 April 2017). Answer to "Banach space that is not a Banach lattice" (accessed 13 August 2022). Mathematics StackExchange. StackOverflow. https://math.stackexchange.com/a/2230649 ↩

3. Schaefer & Wolff 1999, pp. 234–242. - 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. https://search.worldcat.org/oclc/840278135 ↩

4. Birkhoff 1948, p. 251. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. https://hdl.handle.net/2027/iau.31858027322886 ↩

5. Birkhoff 1948, pp. 250, 254. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. https://hdl.handle.net/2027/iau.31858027322886 ↩

6. Birkhoff 1948, pp. 269–271. - Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust. https://hdl.handle.net/2027/iau.31858027322886 ↩