An example of a norm which can't be generated by an inner product

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.

Solution 1:

For example, any $p$-norm except the $2$-norm.

To check this, any norm obtained from the inner-product should satisfy the parallelogram law. Whereas the $p$-norm with $p \neq 2$, does not satisfy the parallelogram law.

Solution 2:

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].