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ring of continuous functions (Definition)

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
    $\displaystyle f^{-1}(x):=\frac{1}{f(x)}.$
    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 $ \vert f\vert(x):=\vert f(x)\vert$, 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+\vert f-g\vert)$. 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:

    $\displaystyle \Vert f\Vert:= \sup_{x \in X} \vert f(x)\vert=\inf \lbrace r\in\mathbb{R} \mid \vert f\vert\le r\rbrace.$
    Routine verifications show that $ \Vert fg\Vert\le \Vert f\Vert\Vert g\Vert$, 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).



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Cross-references: isomorphic, normed ring, norm, compact, pseudocompact, algebraic, easy to see, bounded, subset, lattice-ordered ring, sublattice, join, meet, onto, mapping, operator, absolute value, positive cone, iff, partial order, order, subalgebra, scalar, subring, closed under, invertible, commutative ring, axioms, functional, multiplicative inverse, inverse, additive, multiplicative, constant functions, identities, multiplication, addition, operations, structure, functions, ring, metric topology, reals, continuous functions, function space, topological space
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This is version 11 of ring of continuous functions, born on 2007-04-11, modified 2007-06-08.
Object id is 9176, canonical name is RingOfContinuousFunctions.
Accessed 1261 times total.

Classification:
AMS MSC: 54C35 (General topology :: Maps and general types of spaces defined by maps :: Function spaces)
 54C40 (General topology :: Maps and general types of spaces defined by maps :: Algebraic properties of function spaces)
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