 
Summary: 1. Fourier series.
Definition 1.1. Given a real number P, we say a complex valued function f on R
is Pperiodic if
f(x + P) = f(x) for all x R.
We let
P
be the set of complex valued 2periodic functions f on R such that
1If Leb1 whenever I is a bounded interval.
(Replace Leb1 by Riem1 if Leb1 makes you nervous. A great deal of what follows
will still go through.) It follows from our previous work that P is a vector space
over C with respect to pointwise addition and scalar multiplication.
Here is a Corollary of H¨older's Inequality.
Theorem 1.1. Suppose 1 p < q . Then
fp (2)1/p1/q
fq whenever f P.
In particular,
Pq Pp.
Proof. If q = the inequality holds trivially (Why?) so suppose q < . Let
~p = q/p and ~q = ~p/(~p  1) so ~p and ~q are conjugate. From the H¨older's Inequality
we infer that
