devotion & mathematics

  Multiplicative functions   :              Definition :   ( Multiplicative function )                                                                An arithmetical function f is called multiplicative if f is not identically zero and if f (mn) = f(m)f(n) whenever (m,n) = 1.         Definition :   ( completely multiplicative function )                                       ...

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...