The Stone-Weierstrass Approximation Theorem for Locally Compact Hausdorff Spaces

In this thesis, we present and prove the celebrated Weierstrass Approximation Theorem and a selection of its extensions due to Stone. The final aim is to prove the real and complex versions of the Stone-Weierstrass Approximation Theorem for locally compact Hausdorff spaces, but our ambition is to reach this goal in an intelligible and stepwise manner. Therefore, we first acquaint ourselves with Weierstrass' result concerning polynomial approximation of continuous functions on compact real intervals. With the help of Stone, we then gradually transfer the approximation theory first to arbitrary compact Hausdorff spaces, and eventually to locally compact such.