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.

Cauchy-Schwartz:

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?