ring of continuous functions

Let X be a topological spaceMathworldPlanetmath and C(X) be the function space consisting of all continuous functionsPlanetmathPlanetmath from X into , the reals (with the usual metric topologyMathworldPlanetmath).

Ring Structure on C(X)

To formally define C(X) as a ring, we take a step backward, and look at X, the set of all functions from X to . We will define a ring structureMathworldPlanetmath on X so that C(X) inherits that structure and forms a ring itself.

For any f,gX and any r, we define the following operationsMathworldPlanetmath:

  1. 1.

    (addition) (f+g)(x):=f(x)+g(x),

  2. 2.

    (multiplication) (fg)(x):=f(x)g(x),

  3. 3.

    (identitiesPlanetmathPlanetmathPlanetmath) Define r(x):=r for all xX. These are the constant functions. The special constant functions 1(x) and 0(x) are the multiplicative and additive identities in X.

  4. 4.

    (additive inverse) (-f)(x):=-(f(x)),

  5. 5.

    (multiplicative inverseMathworldPlanetmath) if f(x)0 for all xX, then we may define the multiplicative inverse of f, written f-1 by


    This is not to be confused with the functionalMathworldPlanetmathPlanetmathPlanetmath inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath of f.

All the ring axioms are easily verified. So X 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 0,1C(X), C(X) is a subring of X, and is called the ring of continuous functions over X.

Additional Structures on C(X)

X becomes an -algebra if we define scalar multiplication by (rf)(x):=r(f(x)). As a result, C(X) is a subalgebraMathworldPlanetmath of X.

In addition to having a ring structure, X also has a natural order structure, with the partial order defined by fg iff f(x)g(x) for all xX. The positive conePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is the set {f0f}. The absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath, given by |f|(x):=|f(x)|, is an operator mapping X onto its positive cone. With the absolute value operator defined, we can put a latticeMathworldPlanetmath structure (http://planetmath.org/Lattice) on X as well:

  • (meet) fg:=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) fg:=f+g-(fg).

Since taking the absolute value of a continuous function is again continuous, C(X) is a sublattice of X. 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 boundedPlanetmathPlanetmathPlanetmathPlanetmath 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 compactPlanetmathPlanetmath, C*(X)=C(X).

    In this subring, there is a natural norm that can be defined:


    Routine verifications show that fgfg, so that C*(X) becomes a normed ringMathworldPlanetmath.

  • The subset of C*(X) consisting of all constant functions. This is isomorphicPlanetmathPlanetmathPlanetmath to , and is often identified as such, so that is considered as a lattice-ordered subring of C(X).


  • 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