Proving Schwarz inequaltiy - how to begin

• spaghetti3451
In summary, the conversation discusses solving the Schwarz inequality and using a hint involving bra-ket notation to simplify the problem. The underlying subject of defining an angle between vectors is also mentioned, with the suggestion to use the Cauchy-Schwarz inequality as a guide.
spaghetti3451
Well, I have solved this problem from a textbook:

Prove the Schwarz inequality.

When you try to solve it, you wonder - how do i begin? But thankfully, the problem had a hint: Let $\left|γ\right\rangle = \left|β\right\rangle - \frac{\left\langleα\right| \left|β\right\rangle}{\left\langleα\right| \left|α\right\rangle} \left|α\right\rangle$, and use $\left\langleγ\right| \left|γ\right\rangle \geq 0$.

Well, with this hint, the problem becomes a piece of cake. But how would you know where to start if you didn't have the hint? That's the problem - the starting point. Any ideas?

Last edited:
(The self-admitted voice of ignorance speaking here.) I'm not familiar with bra-ket notation, but see for yourself if this hint is useful to you.

These are vectors in some space, and the underlying subject trying to bubble up here is "how to define what an angle between vectors is". The Cauchy-Schwarz inequality boils down, in simpler contexts, to stating that the cosine of such angle is less than or equal to one. Maybe you can use this (even if just as an analogy) to guide your aim.

1. How do I prove the Schwarz inequality?

The Schwarz inequality can be proven using the Cauchy-Schwarz inequality, which states that for any two vectors a and b, the following inequality holds:
|a·b| ≤ ||a|| · ||b||
Where |a·b| denotes the dot product of a and b, and ||a|| and ||b|| denote the norms of a and b, respectively.

2. What are the steps to prove the Schwarz inequality?

The steps to prove the Schwarz inequality are as follows:
2. Square both sides of the inequality.
3. Expand the squared terms using the distributive property.
4. Rearrange the terms so that all the squared terms are on one side and all the non-squared terms are on the other side.
5. Factor out the squared terms to obtain a perfect square.
6. Take the square root of both sides to obtain the Schwarz inequality.

3. What is the significance of the Schwarz inequality?

The Schwarz inequality is a fundamental inequality in linear algebra that relates the dot product of two vectors to their norms. It has various applications in mathematics and engineering, including optimization problems, signal processing, and quantum mechanics.

4. Can the Schwarz inequality be extended to more than two vectors?

Yes, the Schwarz inequality can be extended to any finite number of vectors. This is known as the generalized Schwarz inequality, which states that for any n vectors a1, a2, ..., an, the following inequality holds:
|a1·a2·...·an| ≤ ||a1|| · ||a2|| ·...·||an||

5. Are there any other inequalities related to the Schwarz inequality?

Yes, there are several other inequalities that are similar to the Schwarz inequality, such as the Holder's inequality and the Minkowski inequality. These inequalities also involve the dot product and norms of vectors and have various applications in mathematics and engineering.

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