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I've just finished a sort of a crash course on functional analysis that left much more questions in me than it answered but one thing bothers me a lot.

When we talk about dual Hilbert spaces it seems that the vector in the dual space has coordinates that are simply complex conjugates of the coordinates of the corresponding vector.

Or using the Dirac notation:

|x>*=<x|, where x is from H - a Hilbert space

|x>=(x

_{1}, x

_{2},...); <x|=(x

_{1}*, x

_{2}*,...)

First I want to ask if I got this right. And second - is there a way to generalise this for dual spaces as a whole (not just Hilbert spaces where H=H*). In other words is there any connection between complex conjugation and dual vectors?