Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Theorem

Let $\struct {K, \tau}$ be a pseudocompact space.

Let $\R$ denote the real number line.

Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.

Let $\struct {\map {C^*} {S, \R}, +, *}$ be the ring of bounded continuous real-valued functions from $S$.

Then:

$\struct {\map {C^*} {S, \R}, +, *} = \struct {\map C {S, \R}, +, *}$

Proof

Follows immediately from the definitions of:

• Definition:Pseudocompact Space

• Definition:Ring of Continuous Real-Valued Functions

• Definition:Ring of Bounded Continuous Real-Valued Functions

$\blacksquare$

Sources

1960: Leonard Gillman and Meyer Jerison: Rings of Continuous Functions: Chapter $1$: Functions on a Topological Space, $\S 1.4$