# Let $V$ be a normed vector space over $\mathbb{C}$, is there an inner product structure on $V$ such that the two spaces have the same topology.

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

• No, it's equivalent. Nov 10, 2013 at 11:34
• @MichaelHoppe Where can I see a proof ?
– Amr
Nov 10, 2013 at 11:38
• 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. Nov 10, 2013 at 11:42
• @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 ?
– Amr
Nov 10, 2013 at 11:46
• @Amr It seems that you question is closely related to the discussion at math.stackexchange.com/questions/540302/… Nov 10, 2013 at 12:05