In mathematical analysis, the Minkowski inequality establishes that the L^p spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of L^p(S). Then f + g is in L^p(S), and we have Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, a measure is a function that assigns a number, e. ...

with equality for 1<p<∞ only if f and g are linearly dependent. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The Minkowski inequality is the triangle inequality in L^p(S). In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure: In mathematics, the counting measure is an intuitive way to put a measure on any set: the size of a subset is taken to be the number of the subsets elements if this is finite, and ∞ if the subset is infinite. ...

for all real (or complex) numbers x₁, ..., x_n, y₁, ..., y_n and where n is the dimension of S. In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Proof

Choose q so that 1/p + 1/q =1. Then

(because |f+g| ≤ |f|+|g|)

(by Hölder's inequality).

Since (p-1)q = p and 1/q = 1-1/p, we obtain Minkowski's inequality by multiplying both sides by ∫(|f+g|^pdμ)^1/p−1. In mathematical analysis, Hölders inequality, named after Otto Hölder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). ...

References

• G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities , Cambridge Univ. Press (1934) ISBN 0-521-35880-9
• H. Minkowski, Geometrie der Zahlen , Chelsea, reprint (1953)
• M.I. Voitsekhovskii, "Minkowski inequality" SpringerLink Encyclopaedia of Mathematics (2001)