Hölder's inequality
In
mathematical analysis,
Hölder's inequality, named after Otto Hölder, is a fundamental inequality relating
Lp 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 L
1(
S) and
By choosing
S to be the set {1,...,
n} with the
counting measure, we obtain as a special case the inequality
valid for all real (or
complex) numbers
x1,...,
xn,
y1,...,
yn.
By choosing
S to be the natural numbers with the counting measure, one obtains a similar 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 Lp and also to establish that Lp is dual to Lq.