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I do not understand how the antilinear correspondence arises between ket vectors in V and bra vectors in V' (dual space):

So I want to know why

$$ \alpha\lt f_1 | + \beta\lt f_2 | $$ corresponds to $$ \alpha^* | f_1 \gt + \beta^* |f_2 \gt $$

where the * represents complex conjugate.

Is there a proof for this?

Ben Gunn
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    Looks like an immediate consequence of a complex inner product being a positive-definite sesquilinear form i.e. $\langle \alpha a, b\rangle = \langle a,\alpha^\ast b\rangle$ – AlexR Mar 11 '15 at 15:01
  • I understand that equation but how does it show a correspondence? Don't I first need to simply show that $\alpha \lt f_1| $ corresponds to $ \alpha^* |f_1 \gt $ (linearity will do the rest). How would I show this then using positive-definite sesquilinear form ? – Ben Gunn Mar 11 '15 at 15:05
  • @AlexR Thank you. Please see here for a proof: http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter9.pdf – Ben Gunn Mar 11 '15 at 18:12
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    They did the same, just avoiding the term. They just call it "complex inner product". – AlexR Mar 11 '15 at 21:24

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