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Hermitian Inner Product

  1. Mar 11, 2006 #1
    I'm getting some confusing information from different sources. If an inner product satisfies conjugate symmetry, it is called Hermitian. But the definition of a hermitian inner product says it must be antilinear in the second slot only. Doesn't conjugate symmetry imply that it's antilinear in both slots?
    Last edited: Mar 11, 2006
  2. jcsd
  3. Mar 11, 2006 #2


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    Conjugate symmetry (plus linearity in the first slot) implies antilinearity in the second:

    [tex]\langle u,\,\alpha v\rangle = \overline{\langle \alpha v,\, u\rangle } = \overline{\alpha \langle v,\, u\rangle } = \overline{\alpha}\overline{\langle v,\, u\rangle } = \overline{\alpha}\langle u,\, v\rangle[/tex]

    If you think conjugate symmetry implies antilinearity in both, present a proof for it.
  4. Mar 11, 2006 #3
    I thought conjugate symmetry and antilinearity in the second slot implied antilinearity in the first, but I made an error when pulling out the constant.
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