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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3You 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 07:39
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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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12And conversely, any norm satisfying the parallelogram law comes from an inner product. This is a nice exercise. – Qiaochu Yuan Jun 18 '12 at 04:04
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6The "nice exercise" that Qiaochu referred to is often called the "polarisation identity". – Willie Wong Jun 18 '12 at 07:52
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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! – information_interchange Mar 21 '20 at 02:46
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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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