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:
- For all reals a1, ... , an, b1, ... , bn,
(a1*b1 + ... + an*bn)2 ≤ (a12 + ... + an2) * (b12 + ... + bn2)
- In an inner product space or a Hilbert space, (x,y) ≤ ||x||*||y||.
- For all random variables X,Y, with standard deviations σ(X), σ(Y), their covariance is bounded by
Cov(X,Y) ≤ σ(X)*σ(Y)
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-norm