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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. $\endgroup$ Jun 18, 2012 at 4:04
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    $\begingroup$ The "nice exercise" that Qiaochu referred to is often called the "polarisation identity". $\endgroup$ Jun 18, 2012 at 7:52
  • $\begingroup$ Wow, this is quite remarkable! I was wondering why it was hard to find which inner product induces L3, L4 etc. norms and we simply have that every $p$ norm except L2 cannot be induced! $\endgroup$ Mar 21, 2020 at 2:46
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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].

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