Which is the Hermitian inner product, in terms of conjugate and transpose?

Question

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$

Page 1 of Source 2: $\mathbf{u \cdot v} = \mathbf{u}^T\mathbf{ \bar{v} }$.

Page 1 of Source 3: Denote $\mathbf{u^*} = \mathbf{\bar{u}^T} $. Then $ \mathbf{ <u,v> = u*v = \bar{u}^Tv } $.

Would someone please explain and elucidate all these differences? Which is right? I'm confused. I believe that $u \cdot v = <u, v>$, if $< >$ is considered as the $\cdot$?

In view of the answer below, which is the most convenient and powerful that I should remember?

Answer 1

Source 1. and 3. are identical. It depends on the convention: there is not really a big difference, just in one case it is linear in the first entry and antilinear ($(u, \alpha v)=\bar\alpha (u,v)$) in the second entry (Source 2) and in the other convention, it is antilinear in the first and linear in the second entry (Source 1 and 3). I believe that more common is the convention from source 1 & 3.

Answer 2

So, as you observed, there are two conventions in the literature for Hermitian inner products:

1. anti-linearity in the first argument, linearity in the second (Sources 1 and 3);
2. linearity in the first argument, anti-linearity in the second (Source 2).

Convention (1) is standard in theoretical physics, probably because of Dirac—I can't remember, off the top of my head, which convention von Neumann uses—and from there in mathematical physics; moreover, it is very common in branches of mathematics traditionally linked to mathematical physics, e.g., operator theory and noncommutative geometry. Convention (2), on the other hand, is common in branches of mathematics where connections to mathematical physics are less prominent.

It should be stressed that there is absolutely no difference in the resulting theory whatsoever from choosing one convention over the other, besides superficial differences in notation here and there, since each convention really is just the complex conjugate of the other.