ring of continuous functions
Let be a topological space and be the function space consisting of all continuous functions from into , the reals (with the usual metric topology).
Ring Structure on
To formally define as a ring, we take a step backward, and look at , the set of all functions from to . We will define a ring structure on so that inherits that structure and forms a ring itself.
For any and any , we define the following operations:
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(addition) ,
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(multiplication) ,
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(identities) Define for all . These are the constant functions. The special constant functions and are the multiplicative and additive identities in .
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(additive inverse) ,
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(multiplicative inverse) if for all , then we may define the multiplicative inverse of , written by
This is not to be confused with the functional inverse of .
All the ring axioms are easily verified. So is a ring, and actually a commutative ring. It is immediate that any constant function other than the additive identity is invertible.
Since is closed under all of the above operations, and that , is a subring of , and is called the ring of continuous functions over .
Additional Structures on
becomes an -algebra if we define scalar multiplication by . As a result, is a subalgebra of .
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:
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(meet) . Here, is the constant function valued at (also as the multiplicative inverse of the constant function ).
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(join) .
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
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, 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.
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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 .
References
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title | ring of continuous functions |
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Canonical name | RingOfContinuousFunctions |
Date of creation | 2013-03-22 16:54:54 |
Last modified on | 2013-03-22 16:54:54 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 14 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54C40 |
Classification | msc 54C35 |