The Supremum Metric
We can actually extend what we’ve been doing with Hölder’s inequality and Minkowski’s inequality a little further. Given a metric space , we’ve already discussed the idea of an “essentially bounded” function — one for which there is some real constant so that for almost all . We will write for the collection of essentially bounded functions on the measure space. It should be clear that these form a vector space.
We also discussed the “essential supremum” of an essentially bounded function. We’ll now write this as , suggesting that it’s a norm. And it’s clear that , and that if and only if almost everywhere. Verifying the triangle identity is exactly Minkowski’s inequality.
And, indeed, we know that and a.e., so a.e., so whatever the least such essential upper bound is smaller still. That is, .
Now for Hölder’s inequality. For this purpose we consider , and thus , which means that and are Hölder-conjugates. Thus our assertion is that if is integrable and is essentially bounded, then is integrable and . Indeed, we know that , and so — both inequalities holding almost everywhere. From this, we conclude that
as we asserted. From now on, we’ll allow (and ) whenever we’re talking about a Hölder-conjugate pair or -space.