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.