Yes you are missing something. In lectures John Jones said that there was an error in the question and that you should replace [a,b] with [0,1].
Let and be the -norm and the -norm on the space of continuous real valued functions on the closed interval : explicitly
Prove that for all (1). Hint: Remember the Cauchy-Schwartz inequality for integrals.
for two real integrable functions in an interval [a,b].
Applying this to I get but this is not what I want to show. Am I missing something here?