What is Minkowski inequality?

In mathematical analysis, the Minkowski inequality establishes that the L spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p < ∞ and let f and g be elements of L. Then f + g is in L, 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. In fact, it is a special case of the more general fact