A Direct Constructive Proof of a Stone-Weierstrass Theorem for Metric Spaces

Abstract

We present a constructive proof of a Stone-Weierstrass theorem for totally bounded metric spaces (\(\mathrm {\mathbf {SWtbms}}\)) which implies Bishop’s Stone-Weierstrass theorem for compact metric spaces (\(\mathrm {\mathbf {BSWcms}}\)) found in [3]. Our proof has a clear computational content, in contrast to Bishop’s highly technical proof of \(\mathrm {\mathbf {BSWcms}}\) and his hard to motivate concept of a (Bishop-)separating set of uniformly continuous functions. All corollaries of \(\mathrm {\mathbf {BSWcms}}\) in [3] are proved directly by \(\mathrm {\mathbf {SWtbms}}\). We work within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\).