MATH 3001 Real Analysis

This is a study of metric spaces, a subject originally initiated by Maurice Fréchet in 1906 and subsequently developed by Hausdorff, Riesz, and ultimately Banach. The original motivation behind the theory idea was to extend the notion of convergence from the real and complex field to more abstract spaces. The idea of the metric was used by Fréchet to induce a topology on the space and later this approach was generalized by Riesz who appears to have been the first to drop the metric idea and replace it by axioms based on sets only. Hausdorff then perfected this study. Banach then introduced normed spaces which can be thought of as special metric spaces where the metric is defined on a vector space.