Proof
First, we prove that f+g has finite p-norm if f and g both do, which follows by
Indeed, here we use the fact that is convex over (for greater than one) and so, by the definition of convexity,
This means that
Now, we can legitimately talk about . If it is zero, then Minkowski's inequality holds. We now assume that is not zero. Using Hölder's inequality
We obtain Minkowski's inequality by multiplying both sides by
Read more about this topic: Minkowski Inequality
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