Norm function

Applicative aspects on functions obtained from generalization concept of norm function

Abstract

The use of norm function in the real world is phenomenal because it is helpful in analyzing the real lengths of the various functions. The purpose of the study is to analyze the different properties and characteristics of the norm function. Through this analysis we obtain that it is not only used in pure mathematics but also significantly used in applied sciences and the major example explained is its use Laplace function which is commonly used in almost every circuit. Langrangian is another use of the norm function which contains different properties that are significantly used in applied sciences.Applicative aspects on functions obtained from generalization concept of norm function

Norm of a vector

Vectors are the measure of the differences of one vector from one another which is called the norm (modulus) of the vector difference. Norm of a vector is determined by many different ways. In theory, all they need to have the following properties:

(1.1)

Here in the first property, the normal (unit) vector X and its multiplication with an absolute value (modulus) of 'a' puts no affect on the vector size as by multiplying it in the modulus. The Third property is called the triangle inequality for the geometric sense (i.e. the length of the triangle does not exceed the sum of the lengths of the other two sides). The second property is held to determine the limit of X sequence of vectors Xn not in a coordinate wise manner (i.e. it coordinates limit vectors defined as the limit of the vectors Xn), and by considering the difference between the normal vectors Xn and the limit vector X:

(1.2)

The most popular ways to determine the norm of a vector are the following two described below:

(1.3)

(1.4)

The norm vector announces most of the modules of its coordinate vector. The Formula |XY| distance between the points X and Y (in columns X, Y can be seen as either necessary or as a point of the radius vector), built on the basis of such a rule is called the uniform metric. Therefore, this rule is called the uniform norm.

The notion of uniform convergence of functions, introduces a mathematical analysis that leads to infinitely continuous function norms. If it is considered that the coordinates of Xn is a function of two arguments i.e. x (k, n), where 'n' is the number of the vector, 'k' is the number of its coordinates, then the convergence of the uniform norm function in (1.3) implies that the convergence of the sequence of functions fn (k) = x(k, n) to the function x(k), uniformly converged with respect to the argument 'k'.

This happens because of the following inequality:

(1.5)

The above inequality can be written as:

(1.6)

It is important to note that in the uniform norm neighborhood of X0 radius R (X points distant from X0 to a distance smaller than R) has the form of n-dimensional cube centered at X 0 and the edges of length 2R, parallel to the coordinate ...