- categories: Real Analysis, Functional analysis, Theorem
Statement
Let be a measure space, and let . For any measurable functions (or ), we have:
For , the inequality holds as well, with:
Intuition
Minkowski’s inequality generalizes the triangle inequality from finite-dimensional spaces to spaces. It says that in an space, the “length” of the sum is at most the sum of the “lengths” of and .
Proof Outline for
- Raise Both Sides to the -th Power: Consider and expand it as:
- Apply the -Norm Triangle Inequality for Sums: Using the convexity of the function (for ) and applying the inequality , we obtain:
- Take the -th Root: Taking the -th root on both sides yields:
Special Cases
- : Minkowski’s inequality reduces to the triangle inequality for integrals:
- : This case corresponds to the triangle inequality in , often used in Hilbert space theory.