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It is known that the norm can induce an inner product if and only if it satisfies Parallelogram law. I just want to know what topology property the inner product has while the norm doesn't have?

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    $\begingroup$ First, do you mean in their corresponding norm topologies? (There are topological differences between some Hilbert and some Banach spaces in their weaker topologies.) Second, probably the best way to pose this question would be "every Hilbert space has property X, does there exist a Banach space which does not have property X?" for particular properties X. $\endgroup$ – Ian Jul 23 '15 at 10:15
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    $\begingroup$ the obvious one: A Hilbert space is isomorphic to its continuous dual, which isn't true for Banach space in general. $\endgroup$ – user251257 Jul 23 '15 at 11:54
  • $\begingroup$ Just to point out, the norm induced by an inner product may well be equivalent to one that is not induced by an inner product - so their topologies might coincide (as they do on $\mathbb{R}^n$, for instance) $\endgroup$ – Prahlad Vaidyanathan Aug 15 '15 at 16:50
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The main difference is that an inner product is a binary operation but norm acts on one element.

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