I am trying to find a counterexample to the parallelogram law to show that the L1 norm is not induced by an inner product in [0,1]. I have found one where I can let f(x) = 1 for x in [0,1] and

g(x) = 1 for x in [1,2]

However, I want one just where the integral has limits 1 and 0 (i.e. just [0,1]).

Thanks.