TITLE: An example of a norm which can't be generated by an inner product
QUESTION [28 upvotes]: I realize that every inner product defined on a vector space can give rise to a norm on that space.  The converse, apparently is not true.  I'd like an example of a norm which no inner product can generate.

REPLY [2 votes]: The norm on C[a,b] defined by ||f||=sup{|f(t)|: t belongs to [a,b]} does not satisfy the parallelogram law. take f(t)=1 and g(t)=t-a/b-a 0r f(t)=max{sin t, 0} and g(t)=max{-sin t, 0) on [0, 2pi].