Editing Bra–ket notation (section)

===Separation of inner product and vectors===
A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as <math>\boldsymbol \psi</math>, and <math>(\cdot,\cdot)</math> for the inner product. Consider the following dual space bra-vector in the basis <math>\{|e_n\rangle\}</math>, where <math>\{\psi_n\}</math> are the complex number coefficients of <math>\langle \psi | </math>:
<math display="block">\langle\psi| = \sum_n \langle e_n| \psi_n</math>

It has to be determined by convention if the complex numbers <math>\{\psi_n\}</math> are inside or outside of the inner product, and each convention gives different results.

<math display="block">\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n</math>
<math display="block">\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n \psi_n, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n^*</math>