We’ve seen the space of integrable functions on a measure space , which we called or . We’ve seen that this gives us a complete normed vector space — a Banach space. This is what we’d like to generalize.
Given a real number , we define the space or to be the collection of all measurable functions for which is integrable. As in the case of , we identify two functions if they’re equal -almost everywhere.
It will turn out that these are Banach spaces. We define the norm
and we write to define a metric. This is clearly non-negative, and we see that if and only if -a.e., just as before. It’s also clear that . What we need to work to check is the triangle inequality. It’s also not quite so apparent a problem, but we actually don’t know yet that this is a vector space at all! That is, how do we know that is integrable if and are?
As a first step in this direction, we prove Hölder’s inequality: if and are real numbers greater than such that , and if and , then the product and . To see this, we will use the function defined for all positive real numbers by
Differentiating, we see that , so the only (positive) critical point of is . Since the limit as approaches and are both positive infinite, must be a local minimum. That is
For any two real numbers and , we can consider the value
and it follows that
which is clearly also true even if we allow or to be zero. This is known as “Young’s inequality”.
Okay, so now we can turn to the theorem itself. If either or , the inequality clearly holds. Otherwise, we define
we can plug these into the above inequality to find
Since the measurability of and implies that of , and the right hand side of this inequality is integrable, we conclude that is integrable. If we integrate, we find
and Hölder’s inequality follows.
The condition relating and is very common in this discussion, so we will say that such a pair of real numbers are “Hölder conjugates” of each other. Given , the Hölder conjugate is uniquely defined by , which is a strictly decreasing function sending to itself (with order reversed, of course). The fact that this function has a (unique) fixed point at will be important. In particular, we will see that this norm is associated with an inner product on , and that Hölder’s inequality actually implies the Cauchy-Schwarz inequality!