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.


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