proof of Cauchy-Schwarz inequality

If a and b are linearly dependent, we write 𝒃=λ𝒂. So we get:


So we have equality if 𝒂 and 𝒃 are linearly dependent. In the other case we look at the quadratic function


This function is positive for every real x, if 𝒂 and 𝒃 are linearly independent. Thus it has no real zeroes, which means that


is always negative. So we have:


which is the Cauchy-Schwarz inequality if 𝒂 and 𝒃 are linearly independent.

Title proof of Cauchy-Schwarz inequality
Canonical name ProofOfCauchySchwarzInequality
Date of creation 2013-03-22 12:34:42
Last modified on 2013-03-22 12:34:42
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 6
Author mathwizard (128)
Entry type Proof
Classification msc 15A63