# Mathematics Prelims

## July 14, 2008

### Minkowski’s Inequality

Filed under: Analysis,Functional Analysis — cjohnson @ 6:58 pm

The final theorem we need before showing the traditional metric on the $l^p$ spaces is indeed a metric is Minkowski’s inequality.

Suppose $(x_n)$ and $(y_n)$ are sequences, $p > 1$ is fixed, and the following series converge.

$\sum_{n=1}^\infty |x_n|^p < \infty$ and $\sum_{n=1}^\infty |y_n|^p < \infty$

Then the following inequality is satisfied.

$(\sum_{n=1}^\infty |x_n + y_n|^p)^{1/p} \leq (\sum_{n=1}^\infty |x_n|^p)^{1/p} + (\sum_{n=1}^\infty |y_n|^p)^{1/p}$.

Proof: Let $z_n = x_n + y_n$, and note $|z_n|^p = |x_n + y_n|^p = |x_n + y_n| |z_n|^{p-1} \leq |x_n| |z_n|^{p-1} + |y_n| |z_n|^{p-1}$.  Summing up to any finite $m$, we have

$\sum_{n=1}^m |z_n|^p \leq \sum_{n=1}^m |x_n| |z_n|^{p-1} + \sum_{n=1}^m |y_n| |z_n|^{p-1}$

Applying Holder’s inequality to each term on the right-hand side, then pulling out the common factor,

$\sum_{n=1}^m |z_n|^p \leq (\sum_{n=1}^m |z_n|^{qp - q})^{1/q} [(\sum_{n=1}^m |x_n|^p)^{1/p} + (\sum_{n=1}^m |y_n|^p)^{1/p}]$

Where $q$ is such that $1/p + 1/q = 1$.  Note that $qp - q = p$.  Dividing through by the factor on the left and noting that $1 - 1/q = 1/p$, we have the desired inequality, but only for finite sums.  (Recall that $|z_n| = |x_n + y_n|$.)

$(\sum_{n=1}^m |z_n|^p)^{1/p} \leq (\sum_{n=1}^m |x_n|^p)^{1/p} + (\sum_{n=1}^m |y_n|^p)^{1/p}$

Now note that this inequality holds for all m, and the series in the terms on the right-hand side converge by assumption, so we can let $m \to \infty$ to obtain the inequality we wanted.