Minkowski inequality

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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 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:

<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.de:Minkowski-Ungleichung

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