When does equality hold? schwarz inequality

[Terrell]

Homework Statement

By choosing the correct vector b in the Schwarz inequality, prove that (a1 + ... + an)^2 =< n(a1^2 + ... +an^2)

Homework Equations

Schwarz inequality

The Attempt at a Solution

since the answer key says that a1 = a2 = ... = an, i tried plugging in values, but i am not getting anywhere and i completely have no intuition of what is happening here. and while on the topic, can anyone recommend books to read - both introductory and more advanced ones about inequalities? i would like a book that gives visual and geometric intuitions. books with proofs would be nice, but it's not a priority to me just yet. thank you!

 

[QuantumQuest]

Homework Statement

By choosing the correct vector b in the Schwarz inequality, prove that (a1 + ... + an)^2 =< n(a1^2 + ... +an^2)

Homework Equations

Schwarz inequality

The Attempt at a Solution

since the answer key says that a1 = a2 = ... = an, i tried plugging in values, but i am not getting anywhere and i completely have no intuition of what is happening here. and while on the topic, can anyone recommend books to read - both introductory and more advanced ones about inequalities? i would like a book that gives visual and geometric intuitions. books with proofs would be nice, but it's not a priority to me just yet. thank you!

You can easily check that for $a_1 = a_2 = \cdots = a_n$ the equality holds. As a hint, use the Cauchy - Schwarz inequality with $b = (1,1,1,..,1)$. You need one more vector a. I leave it to you to figure it out and apply the inequality.

EDIT: I would recommend the book Inequalities by Hardy - Littlewood - Polya https://www.amazon.com/dp/0521358809/?tag=pfamazon01-20. This classic, is a comprehensive study of inequalities. For introductory level, I recommend practicing over a lot of exercises, that can be found easily on the net. But of course, there are lots of good introductory texts too.

 

[Terrell]

You can easily check that for $a_1 = a_2 = \cdots = a_n$ the equality holds. As a hint, use the Cauchy - Schwarz inequality with $b = (1,1,1,..,1)$. You need one more vector a. I leave it to you to figure it out and apply the inequality.

EDIT: I would recommend the book Inequalities by Hardy - Littlewood - Polya https://www.amazon.com/dp/0521358809/?tag=pfamazon01-20. This classic, is a comprehensive study of inequalities. For introductory level, I recommend practicing over a lot of exercises, that can be found easily on the net. But of course, there are lots of good introductory texts too.

thanks i think i got it but i don't get what the "n" in front of "n(a1^2 + ... +an^2)" means

 

[Terrell]

is n = (<1,1,...,1>)^2...?

 

[QuantumQuest]

thanks i think i got it but i don't get what the "n" in front of "n(a1^2 + ... +an^2)" means

In order to understanding it better, give some small value to n and see what the $(a_1 + \cdots + a_n)^{2} \leq n(a_1^{2}+\cdots+a_n^{2})$ gives.

 

[Terrell]

In order to understanding it better, give some small value to n and see what the $(a_1 + \cdots + a_n)^{2} \leq n(a_1^{2}+\cdots+a_n^{2})$ gives.

ahh got it! thanks. that's so simple that it's embarassing