Minkowski inequality
In
mathematical analysis, the
Minkowski inequality establishes that the
Lp 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 Lp(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
x1,...,
xn,
y1,...,
yn.