Minkowski inequality - Definition

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

<math>\|f+g\|_p \le \|f\|_p + \|g\|_p<math>

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:

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

for all real (or complex) numbers x₁, ..., x_n, y₁, ..., y_n.