Inequality that established Lp spaces are normed vector spaces
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.
Proof
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
Minkowski's integral inequality
Suppose that and are two 𝜎-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