Let $X$ be a topological space and $C(X)$ be the function space consisting of all continuous functions from $X$ into $\mathbb{R}$ , 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 $\mathbb{R}^X$ , the set of all functions from $X$ to $\mathbb{R}$ . We will define a ring structure on $\mathbb{R}^X$ so that $C(X)$ inherits that structure and forms a ring itself.

For any $f,g\in \mathbb{R}^X$ and any $r\in\mathbb{R}$ , 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 $x\in X$ . These are the constant functions. The special constant functions $1(x)$ and $0(x)$ are the multiplicative and additive identities in $\mathbb{R}^X$ .
4. (additive inverse) $(-f)(x):=-(f(x))$ ,
5. (multiplicative inverse) if $f(x)\ne 0$ for all $x\in X$ , then we may define the multiplicative inverse of $f$ , written $f^{-1}$ by $$f^{-1}(x):=\frac{1}{f(x)}.$$ This is not to be confused with the functional inverse of $f$ .

All the ring axioms are easily verified. So $\mathbb{R}^X$ 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 $0,1\in C(X)$ , $C(X)$ is a subring of $\mathbb{R}^X$ , and is called the ring of continuous functions over $X$ .

Additional Structures on $C(X)$

$\mathbb{R}^X$ becomes an $\mathbb{R}$ -algebra if we define scalar multiplication by $(rf)(x):=r(f(x))$ . As a result, $C(X)$ is a subalgebra of $\mathbb{R}^X$ .

In addition to having a ring structure, $\mathbb{R}^X$ also has a natural order structure, with the partial order defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$ . The positive cone is the set $\lbrace f\mid 0\le f\rbrace$ . The absolute value, given by $|f|(x):=|f(x)|$ , is an operator mapping $\mathbb{R}^X$ onto its positive cone. With the absolute value operator defined, we can put a lattice structure on $\mathbb{R}^X$ as well:

• (meet) $f\vee g:=2^{-1}(f+g+|f-g|)$ . Here, $2^{-1}$ is the constant function valued at $\frac{1}{2}$ (also as the multiplicative inverse of the constant function $2$ ).
• (join) $f\wedge g:=f+g-(f\vee g)$ .

Since taking the absolute value of a continuous function is again continuous, $C(X)$ is a sublattice of $\mathbb{R}^X$ . 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 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: $$\|f\|:= \sup_{x \in X} |f(x)|=\inf \lbrace r\in\mathbb{R} \mid |f|\le r\rbrace.$$ Routine verifications show that $\|fg\|\le \|f\|\|g\|$ , so that $C^*(X)$ becomes a normed ring.

• The subset of $C^*(X)$ consisting of all constant functions. This is isomorphic to $\mathbb{R}$ , and is often identified as such, so that $\mathbb{R}$ is considered as a lattice-ordered subring of $C(X)$ .

Bibliography

1

L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).