• Source: 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,



≤


{\displaystyle \leq }

, such that for all x, y ∈ X, the implication





|

x

|

≤

|

y

|


⇒

‖
x
‖
≤
‖
y
‖



{\displaystyle {|x|\leq |y|}\Rightarrow {\|x\|\leq \|y\|}}

holds, where the absolute value |·| is defined as




|

x

|

=
x
∨
−
x
:=
sup
{
x
,
−
x
}

.



{\displaystyle |x|=x\vee -x:=\sup\{x,-x\}{\text{.}}}






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



‖
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.




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




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]). The classical mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces.




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.

Kata Kunci Pencarian:

• Banach lattice
• Banach space
• L-space
• List of things named after Stefan Banach
• Normed vector lattice
• Lattice (order)
• Locally convex vector lattice
• Complemented lattice
• Topological vector lattice
• Distributive lattice