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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.

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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.

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    $\begingroup$ And conversely, any norm satisfying the parallelogram law comes from an inner product. This is a nice exercise. $\endgroup$ – Qiaochu Yuan Jun 18 '12 at 4:04
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    $\begingroup$ The "nice exercise" that Qiaochu referred to is often called the "polarisation identity". $\endgroup$ – Willie Wong Jun 18 '12 at 7:52

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