1. Let f ∈ Cc (R) be real-valued and such that the negative and positive parts of f are non-zero.
Show that kf ∗ ∗ f k1 < kf k21 , where f ∗ (x) = f (−x).
2. Using the Hahn-Jordan decomposition of signed measures and the Radon-Nikodym derivative
show that if ν is a complex measure on (X, B) then there exists a positive measure µ and a
measurable function h : X → T such that ν = h µ. The measure µ is usually denoted by |ν|.
3. Let X and Y be compact spaces, π : C(X) → C(Y ) a unital homomorphism. Show that there
exists a continuous map T : Y → X such that π(f ) = f ◦ T for all f ∈ C(X).
1