# Hermitian Inner Product

1. Mar 11, 2006

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. Mar 11, 2006

### AKG

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

$$\langle u,\,\alpha v\rangle = \overline{\langle \alpha v,\, u\rangle } = \overline{\alpha \langle v,\, u\rangle } = \overline{\alpha}\overline{\langle v,\, u\rangle } = \overline{\alpha}\langle u,\, v\rangle$$

If you think conjugate symmetry implies antilinearity in both, present a proof for it.

3. Mar 11, 2006