devotion & mathematics

https://youtu.be/CoYUxNtFOt4?si=1NLs0hEpNkRjJvBo   Contents : 1. Norm of an element 2. Norm of a function 3. Pseudo Metric NORM OF AN ELEMENT :               In each space there is defined a notion of the distance from an arbitrary element to origin, that is, a notion of the " size " of an arbitrary element. The size of an element x is a real number denoted by ||x| | and called its " norm ". For any arbitrary element x , the  norm of x i.e. ||x|| has the following properties :    1. ||x||    ≥0 and          ||x|| =0 iff x=0.    2.  ||-x|| =||x||    3.  ||x+y|| ≤ ||x|| +||y||.             Each metric arises as the norm of the difference between two elements d(x,y)= ||x-y||.      NORM OF A FUNCTION :                     Let X be the set of all bounded continuous real v...

https://youtu.be/lxZV_lBhtaw?si=PFQZ0TVgowSHvhTd 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 :-      ...