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

Suppose and are sequences, is fixed, and the following series converge.

and

Then the following inequality is satisfied.

.

**Proof**: Let , and note . Summing up to any finite , we have

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

Where is such that . Note that . Dividing through by the factor on the left and noting that , we have the desired inequality, but only for finite sums. (Recall that .)

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 to obtain the inequality we wanted.

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