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 ...
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 intr ...
, a Banach lattice is a
complete normed vector space with a
lattice order, 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 1948
N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''.
History
...
of 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 In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X with values in the real or complex numbers. This space, denoted by \mathcal(X), is a vect ...
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 In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive space, reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.
Definition
A uniformly conv ...
, and separable ones are isomorphic to closed sublattices of . The classical
mean ergodic theorem
Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...
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