Hölder’s Inequality
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
and thus
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!