Cauchy-Schwarz Inequality - Duistermaat and Kolk, CH. 1, page 4 .... ....

In summary: Now, we know that <x, y> and <y, y> are both equal to 0, since x and y are orthogonal. This means that the entire expression simplifies to 0, which proves that the two vectors are indeed orthogonal.I hope this helps! Let me know if you have any further questions.
  • #1
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I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of the Cauchy-Schwarz Inequality ...

Duistermaat and Kolk"s proof of the Cauchy-Schwarz Inequality reads as follows:View attachment 7639In the above proof we read the following:

" ... ... Now\(\displaystyle x = \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y + ( x - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y ) \)is a decomposition into two mutually orthogonal vectors ... ... "I have tried to demonstrate that the two vectors \(\displaystyle \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y\) and \(\displaystyle ( x - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y )\) are in fact orthogonal by showing that the inner product of these two vectors is zero ... but I failed to make any meaningful progress ...Can someone please demonstrate that these two vectors are in fact orthogonal ...
Help will be much appreciated ...

Peter
 
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  • #2
Hi Peter,

Set $a=\langle x, y\rangle /\|y\|^{2}$. You want to use the bilinearity property of the (real) inner product to get

$\langle ay, x-ay\rangle = a\langle x, y\rangle - a^{2}\langle y, y\rangle = \langle x, y\rangle ^{2}/\|y\|^{2}-\langle x, y\rangle ^{2}/\|y\|^{2}=0.$

Let me know if anything is unclear.
 
  • #3


Hi Peter,

The proof of the Cauchy-Schwarz Inequality can be a bit tricky to follow, so don't worry if you're having trouble with it. Let me try to explain why the two vectors in question are orthogonal.

First, let's recall the definition of orthogonality. Two vectors x and y are orthogonal if their inner product, denoted by <x, y>, is equal to 0. In other words, if x and y are orthogonal, then <x, y> = 0.

Now, let's look at the two vectors in question. The first vector, \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y, is a multiple of y. This means that it is in the same direction as y, but with a different magnitude. The second vector, (x - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y), is simply the difference between x and the first vector.

If we take the inner product of these two vectors, we get:

<x - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y, \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} y> = \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} <x, y> - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} <y, y>

Since <x, y> is the inner product of two vectors, it is a scalar. Similarly, <y, y> is also a scalar. This means that we can pull these scalars out of the inner product, giving us:

\frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} <x, y> - \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} <y, y> = \frac{ \langle x, y \rangle }{ \mid \mid y \mid \mid^2} \cdot \langle x, y> - \frac{ \langle x, y \rangle
 

Related to Cauchy-Schwarz Inequality - Duistermaat and Kolk, CH. 1, page 4 .... ....

What is the Cauchy-Schwarz Inequality?

The Cauchy-Schwarz Inequality is a mathematical inequality that states the maximum value of the inner product of two vectors is equal to the product of their norms.

Who discovered the Cauchy-Schwarz Inequality?

The Cauchy-Schwarz Inequality is named after mathematicians Augustin-Louis Cauchy and Hermann Schwarz, who both independently discovered it in the 19th century.

What is the significance of the Cauchy-Schwarz Inequality?

The Cauchy-Schwarz Inequality is an important tool in mathematical analysis and has many applications in various fields such as physics, engineering, and economics. It allows us to prove other important theorems and make various calculations.

What is the proof of the Cauchy-Schwarz Inequality?

The Cauchy-Schwarz Inequality can be proven using the Cauchy-Schwarz Inequality Proof, which is based on the concept of orthogonal projection and the Pythagorean Theorem.

How is the Cauchy-Schwarz Inequality used in real life?

The Cauchy-Schwarz Inequality has various applications in real life, such as in statistics to determine correlation between variables, in finance to calculate risk and return, and in machine learning to optimize parameters. It is also used in physics and engineering to solve problems related to vectors and inner products.

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