Minkowski inequality
In
mathematical analysis, the
Minkowski inequality establishes that the
L^{p} spaces are normed vector spaces. Let
S be a
measure space, let 1 ≤
p ≤ ∞ and let
f and
g be elements of L
^{p}(
S). Then
f +
g is in L
^{p}(
S), and we have

with equality only if
f and
g are
linearly dependent.
The Minkowski inequality is the triangle inequality in L^{p}(S). Its proof uses Hölder's inequality.
Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

for all
real (or
complex) numbers
x_{1},...,
x_{n},
y_{1},...,
y_{n}.