We will soon need to know that Hölder’s inequality is in a sense the best we can do, at least for finite . That is, not only do we know that for any and we have , but for any there is some for which we actually have equality. We will actually prove that
That is, not only is the integral bounded above by — and thus by — but there actually exists some in the unit ball which achieves this maximum.
Hölder’s inequality tells us that
so must be at least as big as every element of the given set. If , then it’s clear that the asserted equality holds, since a.e., and so is the only element of the set on the right. Thus from here we can assume .
We now define a function . At every point where we set as well. At all other we define
In the case where we will verify that . That is, the essential supremum of is . And, indeed, we find that at points where , and at points where .
If , then we check
In either case, it’s easy to see that