Is Conjugate Symmetry Enough for a Hermitian Inner Product?

[Treadstone 71]

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?

 

[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.

 

[Treadstone 71]

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.