Also known as the Bunyakovski inequality, particularly in Russian-speaking countries.

This has many forms, but they all involve the comparison of an inner product or a hermitian product with the norms of the components. These forms are particularly useful:

A proof of the Cauchy-Schwarz inequality is elementary, if somewhat tricky.

Hölder's inequality generalizes the Cauchy-Schwarz inequality. A corollary of this inequality is that ||x||=sqrt((x,x)) satisfies the triangle inequality (for norms); the following proof is for real numbers only, but essentially the same proof works for Hilbert space, too.

||x+y||2 = (x+y,x+y) = (x,x)+2(x,y)+(y,y) ≤ (x,x) + 2 sqrt((x,x) * (y,y)) + (y,y) = (||x||+||y||)2
A similar trick with Hölder's inequality gives Minkowski's inequality, for p-norms.