devotion & mathematics

contents : 1. Metric Space. 2. Usual Metric 3. Discrete Metric 4. Indiscrete Metric 5. Metric on set  of real valued continuous functions 6. Metric on  R n  Metric space :- A metric on X is a real function d of ordered pairs (x, y) of elements of X which satisfies the following conditions. (i) d(x, y)≥0 (non- negativity)      and d(x, y)=0 iff x=y. (ii) d(x, y)=d(y, x) (symmetry) (iii) d(x, y) ≤ d(x, z) +d(z, y) (transitivity) Here the space X is called a metric space and it is denoted by (X, d) and the elements of X are called the points of X.     One should always keep in mind, however, that a metric space is not merely a non- empty set: it is a non-empty set together with a metric.          There are many different kinds of metric spaces, some of which play a very significant role in geometry and analysis. A few of them are * Usual metric :-        A function d on the real line R is defined by...

 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....