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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 Lp(S). Then f + g is in Lp(S), and we have

<math>\|f+g\|_p \le \|f\|_p + \|g\|_p</math>
with equality if and only if f and g are linearly dependent[?].

The Minkowksi 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:

<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,...,xn, y1,...,yn.