|
|
|
|
absolute value in a vector lattice
|
(Definition)
|
|
|
Let be a vector lattice over
, and be its positive cone. We define three functions from to as follows. For any ,
It is easy to see that these functions are well-defined. Below are some properties of the three functions:
-
and
.
, since
.
-
, since
.
- If
, then , and . Also, implies , and .
iff . The “only if” part is obvious. For the “if” part, if , then
, so and . But then , so .
-
for any
. If , then
. On the other hand, if , then
.
-
, since
- (triangle inequality).
, since
.
Properties 5, 6, and 8 satisfy the axioms of an absolute value, and therefore is called the absolute value of . However, it is not the “norm” of a vector in the traditional sense, since it is not a real-valued function.
|
"absolute value in a vector lattice" is owned by CWoo.
|
|
(view preamble)
| Also defines: |
absolute value |
This object's parent.
|
|
Cross-references: vector, axioms, triangle inequality, iff, implies, properties, well-defined, easy to see, functions, positive cone, vector lattice
There are 26 references to this entry.
This is version 7 of absolute value in a vector lattice, born on 2007-05-07, modified 2008-06-21.
Object id is 9345, canonical name is AbsoluteValueInAVectorLattice.
Accessed 1410 times total.
Classification:
| AMS MSC: | 06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces) | | | 46A40 (Functional analysis :: Topological linear spaces and related structures :: Ordered topological linear spaces, vector lattices) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|