Cauchy-Varzim inequality is a basic inequality in mathematics, which can be used to solve the maximum and minimum of the inner product of two vectors in vector space.

Let vectors $a$ and $b$ be vectors composed of $n$ real numbers, then their inner products are:

$$a\cdotb=\sum_{i= 1}^na_ib_i$$

Cauchy inequality is expressed as:

$$(a\cdotb)^2\leq(a\cdota)(b\cdotb)$$

The condition of this inequality is that the vectors $a$ and $b$ are not both zero vectors, and the vectors $a$ and $b$ are linearly related.

Cauchy inequality can be used to derive maximum and minimum values in two situations:

1. When the directions of vectors $a$ and $b$ are the same, their inner product is the largest, and the maximum value is $(a\cdota)(b\cdotb)$.

2. When the directions of vectors $a$ and $b$ are opposite, their inner product is the smallest, and the minimum value is $-(a\cdota)(b\cdotb)$.

Cauchy inequality is widely used in mathematics and physics, such as linear algebra, real variable function and so on. Cauchy inequality can be used to simplify the solution process when solving the maximum and minimum values of the inner product.

Introduction to Cauchy:

French mathematician Corcia Augustine Louis (1789- 1857) was born in Paris on August 2/838. His father, Louis Fran? ois Cauchy, was an official of the French Bourbon dynasty and had been holding public office in the turbulent political vortex of France. Due to family reasons, Cauchy himself belongs to the orthodox school that supports the Bourbon dynasty and is a devout Catholic.

His knowledge of pure mathematics and applied mathematics is quite profound, and many mathematical theorems and formulas are named after him, such as Cauchy inequality and Cauchy integral formula. In mathematical writing, he is considered to be second only to Euler in number. He wrote 789 papers and several books in his life, the most famous of which are Analysis Course (182 1 year) and Report on Definite Integral Theory (1827).

However, not all his creations are of high quality, so he was once criticized as "prolific and rash", which is contrary to the Prince of Mathematics (Gauss). It is said that when the Journal of the French Academy of Sciences was first published, there were too many works by Cauchy, and the Academy had to pay a lot of printing expenses, which exceeded the budget of the Academy. So later, the Academy of Sciences stipulated that the longest paper could only reach four pages. Cauchy's longer paper had to be handed in elsewhere.