Which one of the following is the inner product of two complex vectors

Question

I find two different equation in this note and wiki that discuss the linear product of two complex vectors $\langle u,v\rangle$. The note says $\langle u,v\rangle=\overline uv$, whereas wiki says $\langle u,v\rangle=u\overline v$. Furthermore, I've seen both in many places, and I'm wondering which one is correct, or is used more often?

Answer 1

An inner product simply needs to satisfy the inner product axioms, ensuring our vector space is an inner product space. Everything I'm about to say applies to arbitrary IP spaces, including Hilbert spaces.

For real spaces, these axioms include $\langle u,\,v\rangle=\langle v,\,u\rangle$ and linearity in each argument. For complex spaces, we instead have $\langle u,\,v\rangle=\langle v,\,u\rangle^\ast$, and now whichever argument we make linear the other must be antilinear. Thus the inner product becomes sesquilinear instead of bilinear. All this can be reasoned through with words in our head, but sooner or later we have to write down an axiom stating which argument is linear. This is really a matter of convention, since anyone who writes $\langle u,\,v\rangle$ with one preferred answer would write the same complex number as $\langle v,\,u\rangle$ instead, if they later switched conventions.

For whatever (no doubt historical) reasons, we usually say the inner produce is linear in its rightmost argument if we're looking at Hilbert spaces in the context of quantum mechanics, whereas in other contexts mathematicians usually prefer the linearity to be in the leftmost argument. All other axioms are the same either way.