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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 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.

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

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