Hermitian Inner Product

  • #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:

Answers and Replies

  • #2
AKG
Science Advisor
Homework Helper
2,565
4
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.
 
  • #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.
 

Related Threads on Hermitian Inner Product

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
4
Views
922
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
3K
Top