PSEUDO METRIC & NORM OF A FUNCTION #pseudo #metric #norm #of #function
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 valued functions defined on the closed interval [0...