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That is a harder question.. gonna need to sleep on that one.. though @LukeBouck might be able to get it at this hour
In R^n, all norms are equivalent (ie give the same topology) so the inner product would give the same topology as any other norms on R^n.
might be different in infinite dimensional vector spaces. Take L1[0,1]. The L1 and L2 norms aren’t equivalent.
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