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Let $V$ be a complete normed vector space over $\mathbb{C}$. Let $\tau_1$ be the topology induced by its norm. Is there an inner product structure on $V$ such that the topology induced by the norm of the inner product is the same as $\tau_1$ ?

I suspect the answer is no. A sufficent condition for that to happen is that the parallelogram law holds. Is there a weaker known sufficient condition ?

Thank you

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  • $\begingroup$ No, it's equivalent. $\endgroup$
    – Michael Hoppe
    Nov 10, 2013 at 11:34
  • $\begingroup$ @MichaelHoppe Where can I see a proof ? $\endgroup$
    – Amr
    Nov 10, 2013 at 11:38
  • $\begingroup$ Rough sketch: if a norm is induced by an inner product it fulfills the parallelogram identity. If a norm fulfills the identity, then there is only one inner product from where the norm is induced by construction of that inner product from the norm by polarization. $\endgroup$
    – Michael Hoppe
    Nov 10, 2013 at 11:42
  • $\begingroup$ @MichaelHoppe Firstly thanks for your help. Either I misunderstood your argument or you misunderstood my question. I think the last comment you put is an answer to one of my older questions not to this one. The older question is here: math.stackexchange.com/questions/528864/… Agree ? $\endgroup$
    – Amr
    Nov 10, 2013 at 11:46
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    $\begingroup$ @Amr It seems that you question is closely related to the discussion at math.stackexchange.com/questions/540302/… $\endgroup$
    – Alex Ravsky
    Nov 10, 2013 at 12:05

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