On coincidence of algebras of functions with C(X) - Sager, Jürgen (Department of Mathematics and Applied Mathematics, 2016)
The study of approximation of continuous functions has always evoked great interest. The classical result in the theory of approximation is Weierstrass's theorem on the uniform approximation of continuous functions on a bounded closed interval by polynomials. This result was later extended to compact Hausdorff topological spaces by M. H. Stone. Approximation theorems were later also established for almost compact spaces by Hewitt, and for Lindelöf and almost Lindelöf spaces by Hager and Frolik. All the results above were originally proved for real-valued functions. Indeed, they do not generally extend to complex-valued functions; one has to consider different types of algebras and employ different techniques. The theory of complex measures plays an important part in this case. In this thesis we discuss approximation theorems of the Stone-Weierstrass type. We found it convenient to divide the thesis into two parts. Part I concerns real-valued runctions and Part II dicusses the complex case. At the end of each part we have included short bibliographical notes.