Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality relating L^p spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in L^p(S) and g be in L^q(S). Then fg is in L¹(S) and

<math>\|fg\|_1 \le \|f\|_p \|g\|_q.</math>

By choosing S to be the set {1,...,n} with the counting measure, we obtain as a special case the inequality

<math>\sum_{k=1}^n |x_k y_k| \leq \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} \left( \sum_{k=1}^n |y_k|^q \right)^{1/q}</math>

valid for all real (or complex) numbers x₁,...,x_n, y₁,...,y_n. By choosing S to be the natural numbers with the counting measure, one obtains a similiar inequality for infinite series.

For p = q = 2, we get the Cauchy-Schwarz inequality.

Hölder's inequality is used to prove the triangle inequality in the space L^p and also to establish that L^p is dual to L^q.