# Proof of the Cauchy-Shwarz Inequality

• Jow
In summary, the conversation discussed the Cauchy-Schwarz Inequality and an alternative approach to proving it using figure 16. The inequality was shown to be equivalent to the Cauchy-Schwarz Inequality through a series of substitutions and steps. The final result is that ||u•v|| ≤ ||v|| ||u||.
Jow
1. Another approach to the proof of the Cauchy-Shwarz Inequality is suggested by figure 16 (sorry, I don't have the image), which shows that, in ℝ2 or ℝ3, llproj$_u{}$vll ≤ llvll. Show that this inequality is equivalent to the Cauchy-Schwartz Inequality.

2. Cauchy-Schawrtz Inequality: luvl ≤ llull llvll

3. I substituted proj$_u{}$v with (uv/uu)u but I can't think of what to do next.

Last edited:
Actually, I think I got it (please tell me if I am correct):

ll(u•v/u•u)ull ≤ llvll
lu•v/u•ul llull ≤ llvll
lu•v/u•ul^2 llull^2 ≤ llvll^2
lu•v/u•ul lu•v/u•ul (u•u) ≤ v•v
(lu•vl*lu•vl)/(u•u) ≤ v•v
lu•vl^2 ≤ (v•v)(u•u)
lu•vl^2 ≤ llvll^2 llull^2
lu•vl ≤ llvll llull

Jow said:
Actually, I think I got it (please tell me if I am correct):

ll(u•v/u•u)ull ≤ llvll
lu•v/u•ul llull ≤ llvll
lu•v/u•ul^2 llull^2 ≤ llvll^2
lu•v/u•ul lu•v/u•ul (u•u) ≤ v•v
(lu•vl*lu•vl)/(u•u) ≤ v•v
lu•vl^2 ≤ (v•v)(u•u)
lu•vl^2 ≤ llvll^2 llull^2
lu•vl ≤ llvll llull

That's pretty hard to read. But it look ok to me. There's probably some extra steps in there you don't need. u.u=||u||^2, right?

Yeah, I put few extra steps that weren't necessary. I know its difficult to read; I am not really familiar with how to properly input some of the symbols.

Jow said:
ll(u•v/u•u)ull ≤ llvll

Jow said:
Yeah, I put few extra steps that weren't necessary. I know its difficult to read; I am not really familiar with how to properly input some of the symbols.

It isn't that difficult

##\| (\frac{u\cdot v}{u\cdot u}) u \| \le \| v \|##
Right click on it to see the code. Put ## at the beginning and end of the code.

## 1. What is the Cauchy-Shwarz Inequality?

The Cauchy-Shwarz Inequality, also known as the Cauchy-Bunyakovsky-Schwarz Inequality, is a mathematical inequality that relates to the inner product of two vectors in a vector space. It states that the absolute value of the inner product of two vectors is less than or equal to the product of the magnitudes of the two vectors, and that equality is only achieved when the vectors are linearly dependent.

## 2. Who discovered the Cauchy-Shwarz Inequality?

The Cauchy-Shwarz Inequality was discovered and proven by Augustin-Louis Cauchy, a French mathematician, in the early 19th century. However, it was later generalized and popularized by Viktor Bunyakovsky and Hermann Schwarz, hence the alternate names of the inequality.

## 3. What are the applications of the Cauchy-Shwarz Inequality?

The Cauchy-Shwarz Inequality has numerous applications in mathematics, physics, and engineering. It is used in the proof of many other mathematical theorems, such as the Triangle Inequality and the Hölder's Inequality. It also has applications in statistics, signal processing, and optimization problems.

## 4. How is the Cauchy-Shwarz Inequality proved?

The Cauchy-Shwarz Inequality can be proved using various methods, including the algebraic method, the geometric method, and the linear algebra method. The algebraic method involves manipulating the inner product and showing that it is less than or equal to the product of the magnitudes of the two vectors. The geometric method involves using the concept of orthogonal projections to prove the inequality. The linear algebra method involves using properties of matrices and determinants to prove the inequality.

## 5. Are there any variations of the Cauchy-Shwarz Inequality?

Yes, there are several variations of the Cauchy-Shwarz Inequality, including the Cauchy-Davenport Inequality, the Minkowski Inequality, and the Hölder's Inequality. These variations have different forms and are applicable in different scenarios, but they are all based on the same fundamental concept of the Cauchy-Shwarz Inequality.

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