Matlab to solve the norm of linear equation group

1. normed linear space and inner product space

In the primary textbook of linear algebra, the inner product is usually defined in the vector space, and then the norm is derived from the inner product, such as in the n-dimensional real vector space:

|x| | =√<x,x>

In the advanced textbook of linear algebra, the inner product and norm are generally defined separately, and there may not be a direct relationship between them. By introducing the norm into the vector space, we can get an normed linear space (normed linear spaces) and introduce the inner product in the vector space, and we can get an inner product space (inner product spaces). The definition is as follows:

properties of the inner product space

Cauchy-schwarz Inequalities

The normed linear space produced by the norm derived from the inner product

Given an inner product space, you can define a norm: | |x| | =<x,x>½ generally call this norm the norm derived from the inner product. It can be found that if the inner product space is added to a norm derived from the inner product, then the inner product space is an normed linear space. This normed linear space has the following special properties, which are not available in ordinary normed linear spaces:

A few common norm

Analytic properties of Norm

As seen above, in vector space V, there are multiple real-valued functions to satisfy the norm Axiom, that is, there are multiple norms, a very natural question is, what kind of relationship between these norms, there are two important theorems: norm equivalence and completeness.

The norm in a finite-dimensional normed linear space is equivalent:

The finite dimensional normed linear space is complete:

Matlab to solve the norm of linear equation group