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 ${ℝ}^X$, the set of all functions from $X$ to $ℝ$. We will define a ring structure on ${ℝ}^X$ so that $C(X)$ inherits that structure and forms a ring itself.

For any $f,g\in {ℝ}^X$ and any $r\in ℝ$, 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 ${ℝ}^X$.

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 ${ℝ}^X$ 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 ${ℝ}^X$, and is called the ring of continuous functions over $X$.

Additional Structures on $C(X)$

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

In addition to having a ring structure, ${ℝ}^X$ 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 $\{f\mid 0\le f\}$. The absolute value, given by $|f|(x):=|f(x)|$, is an operator mapping ${ℝ}^X$ onto its positive cone. With the absolute value operator defined, we can put a lattice structure (http://planetmath.org/Lattice) on ${ℝ}^X$ 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 ${ℝ}^X$. 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:

  $$\parallel f\parallel :=\underset{x\in X}{sup}|f(x)|=inf\{r\in ℝ\mid |f|\le r\}.$$

  Routine verifications show that $\parallel fg\parallel \le \parallel f\parallel \parallel g\parallel$, 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 $C(X)$.