absolute value in a vector lattice
Let be a vector lattice over , and be its positive cone. We define three functions from to as follows. For any ,
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,
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,
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.
It is easy to see that these functions are well-defined. Below are some properties of the three functions:
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1.
and .
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2.
, since .
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3.
, since .
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4.
If , then , and . Also, implies , and .
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5.
iff . The “only if” part is obvious. For the “if” part, if , then , so and . But then , so .
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6.
for any . If , then . On the other hand, if , then .
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7.
, since
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8.
(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.
Title | absolute value in a vector lattice |
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Canonical name | AbsoluteValueInAVectorLattice |
Date of creation | 2013-03-22 17:03:16 |
Last modified on | 2013-03-22 17:03:16 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 10 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 46A40 |
Classification | msc 06F20 |
Defines | absolute value |