# Schwarz Inequality Proof

1. Sep 14, 2008

### Old Guy

1. The problem statement, all variables and given/known data

2. Relevant equations
[itex](\langle \alpha | + \lambda^\ast\langle \beta |)(|\alpha\rangle+ \lambda|\beta\rangle) = \langle \alpha |\alpha \rangle + |\lambda|^2\langle \beta | \beta \rangle + \lambda \langle \alpha | \beta \rangle + \lambda^\ast \langle \beta | \alpha \rangle \geq 0[itex]

3. The attempt at a solution
This equation is the setup, and it leads to an equation that I can see is quadratic in lambda. From this, I calculate the discriminant, which must be greater than or equal to zero because all the terms are real and positive. However, when I manipulate this to get to the Schwarz inequality, I get a "less than or equal to" where I should have a "greater than or equal to". Can somone please help? Thanks.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 14, 2008

### Dick

Use tex and /tex for the tex delimiters instead of latex and latex. And you don't want to solve a quadratic. Just put in the special value lambda=-<beta|alpha>/<beta|beta>.

Last edited: Sep 14, 2008
3. Sep 14, 2008

### Old Guy

Thanks, and sorry about the equation format; I'm still trying to figure out the MathType translator.

Anyway, I understand how it works for the value of lambda you gave, but shouldn't it work for ANY value of lambda?

4. Sep 14, 2008

### Dick

It works for any lambda, but it doesn't tell you anything very interesting for every lambda. E.g. lambda=0 isn't interesting at all.