The supremum norm on C[a,b] can not be obtained from an inner product. Since if f(t) = 1 and g(t) = [(t-a)/(b-a)], then ||f || = ||g || = 1, ||f-g || = sup{ |1 - [(t-a)/(b-a)] | ; t Î [a,b] } = 1 and ||f+g || = sup{ |1 + [(t-a)/(b-a)] | ; t Î [a,b] } = 2 and so the parallelogram equality ||f+g ||² + ||f-g ||² = 2 ||f ||² + 2 ||g ||² (which is satisfied in every inner product space) isn't held.

Comment. Indeed this Banach space is not an inner product space in any equivalent norm.