# Homework Help: How to prove Schwartz Inequality?

1. Mar 14, 2009

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

Prove the Schwartz Inequality

/<u, v>/ is < or = //u//*//v//

for the Hermitian product.

2. Relevant equations

3. The attempt at a solution

Assume u, v are non-zero. For any r, s exist in set of complex numbers, //rv + su// is > or = 0. Then, //rv + su//^2 = <rv+su, rv+su> = <rv, rv> + <rv, su> + <su, rv> + <su, su> = r^2<v, v> + rs <v, u> + sr <u, v> + s^2 <u, u> is > or = 0

Not sure where to go from there...set r = <u u> and s = ....?? I don't know.

Note: I was working off of the proof of the inequality for the dot product. I am just not sure how the Hermitian properties affect it. is <v, u> = <u, v>, etc.

2. Mar 14, 2009

### roam

We say (u.v)2 ≤ |u|2 |v|2, then if u or v = 0 then both sides of both sides are zero. But if they are not zero, the vector v can be rewritten as the sum of some scalar multiple of u, ku, and a vector w that is orthogonal to u. The k is obtained by w = v-ku & the orthogonality condition u.w=0

0 = u.w = u.(v-ku) = (u.v)-k(u.u), it follows that k = u.v/u.u, now apply the pythagoras theorem:

|v|2 = |ku|2 + |w|2 = k|u|2 + |w|2

|u|2|v|2 = (u.v) + |u|2|w|2

since |u|2|w|2 ≥ 0, you have (u.v)2 ≤ |u|2 |v|2
This establishes what we said first and hence the cauchy shwartz inequality.