HISTORICAL BACKGROUND OF METRIC SPACES : #historical #backgrond #metric #spaces #convergence #continuous
HISTORICAL BACKGROUND OF METRIC SPACES:
Classical analysis can be described as that part of mathematics which begins with calculus and, in essentially the same spirit, develops similar subject matter much further in many directions. It is a great nation in the world of mathematics, with many provinces, a few of which are ordinary and partial differential equations, infinite series, and analytic functions of a complex variable. Each of these has experienced enormous growth over a long history, and each is rich enough in content to merit a lifetime of study.
In the course of its development, classical analysis became so complex and varied that even an expert could find his way around in it only with difficulty. Under these circumstances, some mathematicians became interested in trying to uncover the fundamental principles on which all analysis rests. This movement had associated with it many of the great names in mathematics of the last century : Riemann, Weierstrass, Cantor, Lebesgue, Hilbert , Riesz and others. It played a large part in the rise to prominence of topology, modern algebra, and the theory of measure and integration: and when these new ideas began to percolate back through classical analysis, the brew which resulted was modern analysis.
As modern analysis developed in the hands of its creators, many a major theorem was given a simpler proof in a more general setting, in an effort to lay bare its inner meaning. Much thought was devoted to analyzing he texture of the real and complex number systems, which are the context of the analysis. It was hoped- and these hopes were well founded that analysis could be clarified and simplified.
Analysis is primarily concerned with limit processes and continuity, so it is not surprising that mathematicians thinking along these lines soon found themselves two elementary concepts: that of a convergent of a sequence of real or complex numbers, and that of continuous function of real or complex variable.
Our purpose in giving the definitions of convergence of a sequence and continuity of a function in detail here is a simple one. We wish to point out explicitly that each is dependent for its meaning on the concept of the absolute value of the difference between two real or complex numbers. We wish to observe also that this absolute value is the distance between the numbers when they are regarded as points on the real line or complex plane.
In many branches of mathematics - in geometry as well as analysis- it has been found extremely convenient to have available a notion of distance which is applicable to the elements of abstract sets. A metric space is nothing more than a nonempty set equipped with a concept of distance which is suitable for the treatment of convergent sequences in the set and continuous functions defined on the set. Our purpose is to develop in a symmetric manner the main elementary facts about metric spaces. These facts are important for their own sake, and also for the sake of the motivation they provide for our work on topological spaces.
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#historical #backgrond #metric #spaces #convergence #continuous
Nice introduction
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