# 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.

-
You may be interested in having a look at the the threads: "Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)" (containing a slightly more advanced perspective on your question) and "Norms Induced by Inner Products and the Parallelogram Law" (an outline and a detailed solution to the exercise "if a norm satisfies the parallelogram law then it's an inner product" suggested by Qiaochu in the comments). –  t.b. Jun 18 '12 at 7:39

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.