Analysis of Metric Spaces

Metric spaces were introduced and studied by the French mathematician, Maurice René Fréchet (in his doctoral dissertation published in 1906), and developed later by the German Felix Hausdorff (in his 1914 book Grundzüge der Mengenlehre). It was apparent that to the end of the nineteenth century the mathematical world (partly inspired by Cantor’s fundamental work in set theory) was eager to structure more general sets than conventional ℝn. On the other hand, the needs of complex analysis and the rash development of differential equations accelerated this process. Typical examples are uniform convergence in function spaces, approximation of continuous functions by polynomials and the Riemann mapping theorem. After 1920, the theory of metric spaces, especially fundamental work on normed spaces and their applications to functional analysis, was further developed by the Pole Stefan Banach and his school. Paying tribute to their achievements and of their fellow countrymen followers, an important subclass of metric spaces was named “Polish.” A series of studies of metric spaces was further undertaken in the late 1920s by the Russian school of analysis. At this time, metric spaces have become generalized to topological spaces.