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 the triangle inequality

with equality for 1 < p < ∞ if and only if f and g are positively linearly dependent, i.e., f = λg for some λ ≥ 0 or g = 0. Here, the norm is given by:

if p < ∞, or in the case p = ∞ by the essential supremum

The Minkowski inequality is the triangle inequality in L^p(S). In fact, it is a special case of the more general fact

where it is easy to see that the right-hand side satisfies the triangular 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, ..., x_n, y₁, ..., y_n and where n is the cardinality of S (the number of elements in S).

First, we prove that f+g has finite p-norm if f and g both do, which follows by

Indeed, here we use the fact that ${\displaystyle h(x)=x^{p}}$ is convex over R⁺ (for p > 1) and so, by the definition of convexity,

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