How to prove Schwartz Inequality?

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Homework Statement

Prove the Schwartz Inequality

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

for the Hermitian product.

Homework Equations

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.

 

[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.