ring of continuous functions
Ring Structure on
For any and any , we define the following operations:
Additional Structures on
In addition to having a ring structure, also has a natural order structure, with the partial order defined by iff for all . The positive cone is the set . The absolute value, given by , is an operator mapping onto its positive cone. With the absolute value operator defined, we can put a lattice structure (http://planetmath.org/Lattice) on as well:
(meet) . Here, is the constant function valued at (also as the multiplicative inverse of the constant function ).
Since taking the absolute value of a continuous function is again continuous, is a sublattice of . As a result, we may consider as a lattice-ordered ring of continuous functions.
Remarks. Any subring of is called a ring of continuous functions over . This subring may or may not be a sublattice of . Other than , the two commonly used lattice-ordered subrings of are
, the subset of consisting of all bounded continuous functions. It is easy to see that is closed under all of the algebraic operations (ring-theoretic or lattice-theoretic). So is a lattice-ordered subring of . When is pseudocompact, and in particular, when is compact, .
In this subring, there is a natural norm that can be defined:
Routine verifications show that , so that becomes a normed ring.
The subset of consisting of all constant functions. This is isomorphic to , and is often identified as such, so that is considered as a lattice-ordered subring of .
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
|Title||ring of continuous functions|
|Date of creation||2013-03-22 16:54:54|
|Last modified on||2013-03-22 16:54:54|
|Last modified by||CWoo (3771)|