In the
mathematical disciplines of in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
and
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
, a Banach lattice is a
complete normed vector space with a
lattice order
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
, such that for all , 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
n 1948of a Banach space
asalso a vector lattice." In particular:
* , together with its absolute value as a norm, is a Banach lattice.
* Let be a topological space, a Banach lattice and the
space of continuous bounded functions from to with norm
Then is a Banach lattice under the pointwise partial order:
Examples of non-lattice Banach spaces are now known;
James' space is one such.
[Kania, Tomasz (12 April 2017).]
Answer
to "Banach space that is not a Banach lattice" (accessed 13 August 2022). ''Mathematics StackExchange''. StackOverflow
In software, a stack overflow occurs if the call stack pointer exceeds the stack bound. The call stack may consist of a limited amount of address space, often determined at the start of the program. The size of the call stack depends on many facto ...
.
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
is called an abstract (L)-space. Such spaces are necessarily
uniformly convex, and separable ones are isomorphic to closed sublattices of . The classical
mean ergodic theorem and
Poincaré recurrence
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luci ...
generalize to abstract (L)-spaces.
See also
*
*
*
*
Footnotes
Bibliography
*
*
*
*
{{mathanalysis-stub
Functional analysis
Order theory