Minkowski inequality

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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

<math>\|f+g\|_p \le \|f\|_p + \|g\|_p</math>

with equality only if f and g are linearly dependent.

The Minkowski inequality is the triangle inequality in L^p(S). Its proof uses Hölder's inequality.

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

<math>\left( \sum_{k=1}^n |x_k + y_k|^p \right)^{1/p} \le \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} + \left( \sum_{k=1}^n |y_k|^p \right)^{1/p}</math>

for all real (or complex) numbers x₁, ..., x_n, y₁, ..., y_n and where n is the dimension of S.

Proof of Integral form

<math>\left(\int_{a}^{b}|f(x)+g(x)|^{p}dx\right)^{\frac{1}{p}*p}=\int_{a}^{b}|f(x)+g(x)||f(x)+g(x)|^{p-1}dx</math>

Expanding <math>|f(x)+g(x)|</math>

<math>\leq\int_{a}^{b}|f(x)||f(x)+g(x)|^{p-1}dx+\int_{a}^{b}|g(x)||f(x)+g(x)|^{p-1}dx</math>

Then using Hölder's inequality

<math>\leq\left(\int_{a}^{b}|f(x)|^{p}dx\right)^{\frac{1}{p}}\left(\int_{a}^{b}|f(x)+g(x)|^{q\left(p-1\right)}dx\right)^{\frac{1}{q}}+ \left(\int_{a}^{b}|g(x)|^{p}dx\right)^{\frac{1}{p}}\left(\int_{a}^{b}|f(x)+g(x)|^{q\left(p-1\right)}dx\right)^{\frac{1}{q}}</math>

Simplified

<math>\leq\left[\left(\int_{a}^{b}|f(x)|^{p}dx\right)^{\frac{1}{p}}+\left(\int_{a}^{b}|g(x)|^{p}dx\right)^{\frac{1}{p}}\right]\left(\int_{a}^{b}|f(x)+g(x)|^{qp-q}dx\right)^{\frac{1}{q}}</math>

As <math>\frac{1}{p}+\frac{1}{q}=1</math>;

<math>\frac{1}{q}=1-\frac{1}{p}</math>, <math>1=q-\frac{q}{p}</math>, <math>p=qp-q</math>,

Therefore

<math>\leq\left[\left(\int_{a}^{b}|f(x)|^{p}dx\right)^{\frac{1}{p}}+\left(\int_{a}^{b}|g(x)|^{p}dx\right)^{\frac{1}{p}}\right]\left(\int_{a}^{b}|f(x)+g(x)|^{p}dx\right)^{\frac{1}{p}*\frac{p}{q}}</math>

Dividing by the right integral

<math>\left(\int_{a}^{b}|f(x)+g(x)|^{p}dx\right)^{\frac{1}{p}\left(p-\frac{p}{q}\right)}=\left[\left(\int_{a}^{b}|f(x)|^{p}dx\right)^{\frac{1}{p}}+\left(\int_{a}^{b}|g(x)|^{p}dx\right)^{\frac{1}{p}}\right]</math>

And as <math>p-\frac{p}{q}=1</math>

<math>\left(\int_{a}^{b}|f(x)+g(x)|^{p}dx\right)^{\frac{1}{p}}=\left[\left(\int_{a}^{b}|f(x)|^{p}dx\right)^{\frac{1}{p}}+\left(\int_{a}^{b}|g(x)|^{p}dx\right)^{\frac{1}{p}}\right]</math>de:Minkowski-Ungleichung ru:Неравенство Минковского tr:Minkowski Eşitsizliği