ordered vector space
for any , if then ,
if and any , then .
Here is a property that can be immediately verified: iff for any .
Also, note that is interpreted as the zero vector of , not the bottom element of the poset . In fact, is both topless and bottomless: for if is the bottom of , then , or , which implies or . This means that for all . But if , then or , a contradiction. is topless follows from the implication that if exists, then is the top.
For example, any finite dimensional vector space over , and more generally, any (vector) space of real-valued functions on a given set , is an ordered vector space. The natural ordering is defined by iff for every .
Properties. Let be an ordered vector space and . Suppose exists. Then
exists and for any vector .
Let . Then and . For any upper bound of and , we have and . So , or . So is the least upper bound of and . ∎
exists and .
Let . Since , , so . Similarly , so is a lower bound of and . If and , then and , or and , or , or . Hence the greatest lower bound of and . ∎
exists for any scalar , and
if , then
if , then
if , then the converse holds for (a) and (b).
Assume (clear otherwise). (a). If , implies . Similarly, . If and , then and , hence , or . Proof of (b) is similar to (a). (c). Suppose and . Set . Then . This implies , or , a contradiction. ∎
A vector space over is said to be ordered if is an ordered vector space over , where ( is the complexification of ).
Given any vector space, a proper cone defiens a partial ordering on , given by if . It is not hard to see that the partial ordering so defined makes into an ordered vector space.
So, there is a one-to-one correspondence between proper cones of and partial orderings on making an ordered vector space.
|Title||ordered vector space|
|Date of creation||2013-03-22 16:37:24|
|Last modified on||2013-03-22 16:37:24|
|Last modified by||CWoo (3771)|
|Synonym||ordered linear space|