Minkowski inequality

The Minkowski inequality ( after Hermann Minkowski ) is a statement of functional analysis. It says that the triangle inequality is true in the Lp- spaces.

Formulation

Let S be a measure space, as well. Then follows, and it is

Where the equality in the case if and only exists when f and g are linearly dependent positive (ie there with or ).

This is the standard by

Given if the measure referred to.

Evidence

The Minkowski inequality is trivial for and. It is therefore. As a convex function is valid

And therefore.

Be in the following without loss of generality. The following applies:

Be. Then q is the Hölder exponent conjugate to p, we have:

After the Hölder inequality holds:

This implies the Minkowski inequality by multiplying both sides by.

Special case

How Hölder's inequality can also be the Minkowski inequality to consequences ( in the first Example below: finite sequences, ie, n- tuples of real ( or complex ) entries) are specialized by using the counting measure:

For all real ( or complex ) numbers. The Minkowski inequality is thus the triangle inequality for the p- norms. Generally, one can, for infinite sequences, even

. Write (This is always: Because if one of the two sums on the right hand diverges, so the inequality holds because then for all. )

Generalization ( Minkowski 's inequality for integrals)

Let and be two measurable spaces and measurable function, then applies ( Minkowski 's inequality for integrals):

For. If both sides and is finite, then equality holds if and only if can be a product of two measurable functions and write.

If we choose as the two - element set with the counting measure, we obtain as a special case again the usual Minkowski inequality, with for is namely