ring of continuous functions
Let X be a topological space and C(X) be the function space consisting of all continuous functions
from X into ℝ, the reals (with the usual metric topology
).
Ring Structure on C(X)
To formally define C(X) as a ring, we take a step backward, and look at ℝX, the set of all functions from X to ℝ. We will define a ring structure on ℝX so that C(X) inherits that structure and forms a ring itself.
For any f,g∈ℝX and any r∈ℝ, we define the following operations:
-
1.
(addition) (f+g)(x):=f(x)+g(x),
-
2.
(multiplication) (fg)(x):=f(x)g(x),
-
3.
(identities
) Define r(x):=r for all x∈X. These are the constant functions. The special constant functions 1(x) and 0(x) are the multiplicative and additive identities in ℝX.
-
4.
(additive inverse) (-f)(x):=-(f(x)),
-
5.
(multiplicative inverse
) if f(x)≠0 for all x∈X, then we may define the multiplicative inverse of f, written f-1 by
f-1(x):=1f(x). This is not to be confused with the functional
inverse
of f.
All the ring axioms are easily verified. So ℝX is a ring, and actually a commutative ring. It is immediate that any constant function other than the additive identity is invertible.
Since C(X) is closed under all of the above operations, and that 0,1∈C(X), C(X) is a subring of ℝX, and is called the ring of continuous functions over X.
Additional Structures on C(X)
ℝX becomes an ℝ-algebra if we define scalar multiplication by (rf)(x):=r(f(x)). As a result, C(X) is a subalgebra of ℝX.
In addition to having a ring structure, ℝX also has a natural order structure, with the partial order defined by f≤g iff f(x)≤g(x) for all x∈X. The positive cone is the set {f∣0≤f}. The absolute value
, given by |f|(x):=|f(x)|, is an operator mapping ℝX onto its positive cone. With the absolute value operator defined, we can put a lattice
structure (http://planetmath.org/Lattice) on ℝX as well:
-
•
(meet) f∨g:=2-1(f+g+|f-g|). Here, 2-1 is the constant function valued at 12 (also as the multiplicative inverse of the constant function 2).
-
•
(join) f∧g:=f+g-(f∨g).
Since taking the absolute value of a continuous function is again continuous, C(X) is a sublattice of ℝX. As a result, we may consider C(X) as a lattice-ordered ring of continuous functions.
Remarks. Any subring of C(X) is called a ring of continuous functions over X. This subring may or may not be a sublattice of C(X). Other than C(X), the two commonly used lattice-ordered subrings of C(X) are
-
•
C*(X), the subset of C(X) consisting of all bounded
continuous functions. It is easy to see that C*(X) is closed under all of the algebraic operations (ring-theoretic or lattice-theoretic). So C*(X) is a lattice-ordered subring of C(X). When X is pseudocompact, and in particular, when X is compact
, C*(X)=C(X).
In this subring, there is a natural norm that can be defined:
∥f∥:=supx∈X|f(x)|=inf{r∈ℝ∣|f|≤r}. Routine verifications show that ∥fg∥≤∥f∥∥g∥, so that C*(X) becomes a normed ring
.
-
•
The subset of C*(X) consisting of all constant functions. This is isomorphic
to ℝ, and is often identified as such, so that ℝ is considered as a lattice-ordered subring of C(X).
References
- 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Title | ring of continuous functions |
---|---|
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 |