Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let be a measure space, let and let and be elements of Then is in and we have the triangle inequality with equality for if and only if and are positively linearly dependent; that is, for some or Here, the norm is given by: if or in the case by the essential supremum The Minkowski inequality is the triangle inequality in 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 and where is the cardinality of (the number of elements in ). The inequality is named after the German mathematician Hermann Minkowski. First, we prove that has finite -norm if and both do, which follows by Indeed, here we use the fact that is convex over (for ) and so, by the definition of convexity, This means that Now, we can legitimately talk about If it is zero, then Minkowski's inequality holds. We now assume that is not zero. Using the triangle inequality and then Hölder's inequality, we find that We obtain Minkowski's inequality by multiplying both sides by Suppose that and are two sigma-finite measure spaces and is measurable. Then Minkowski's integral inequality is , : with obvious modifications in the case If and both sides are finite, then equality holds only if a.e. for some non-negative measurable functions and If is the counting measure on a two-point set then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting for the integral inequality gives If the measurable function is non-negative then for all This notation has been generalized to for with Using this notation, manipulation of the exponents reveals that, if then When the reverse inequality holds: We further need the restriction that both and are non-negative, as we can see from the example and The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range.