Banach lattice

In the <a href="/facts/Mathematics/pxTouaz4">mathematical disciplines</a> of in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, a Banach lattice (X,‖·‖) is a <a href="/facts/Banach_space/sDEIk7EC">complete normed vector space</a> with a <a href="/facts/Lattice_order/5ak6ufky">lattice order</a>, 
 
 
 
 ≤
 
 
 {\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{.}}}

Banach lattice open-in-new

Banach lattice