Analysis of Metric Spaces
摘要
Metric spaces were introduced and studied by the French mathematician, \nMaurice Renk Frkchet (in his doctoral dissertation published in 1906), \nand developed later by German Felix Hausdorff (in his book Grundziige \nder Mengenlehre of 1914). It was apparent that to the end of the nineteenth century the mathematical world (partly inspired by Cantor&s;s \nfundamental work in set theory) was eager to structure more general sets \nthan conventional Rn. On the other hand, the needs of complex analysis \nand the rash development of differential equations speeded up this \nprocess. Typical examples are uniform convergence in function spaces, \napproximation of continuous functions by polynomials and the Riemann \nmapping theorem. After 1920, the theory of metric spaces, especially, \nfundamental work on normed spaces and their applications to functional \nanalysis, was further developed by Pole Stefan Banach and his school. \nPaying a tribute to their achievements and of their other fellow countrymen followers, an important subclass of metric spaces was named \n"Polish." A series of studies of metric spaces were further undertaken in \nthe late 1920s by the Russian school of analysis. At this time, metric \nspaces have become generalized to topological spaces.