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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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    You may be interested in havin$g$ a look at the the threads: "[Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)](http://math.stackexchange.com/q/38460/5363)" (containing a slightly more advanced perspective on your question) and "[Norms Induced by Inner Products and the Parallelogram Law](http://math.stackexchange.com/q/21792/5363)" (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).2012-06-18

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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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    The "nice exercise" that Qiaochu referred to is often called the "[polarisation identity](http://en.wikipedia.org/wiki/Polarization_identity)".2012-06-18