For a sesquilinear form, I have encountered two sources that say completely opposite things.
During my lecture, my teacher said $\langle \alpha v|w\rangle=\alpha^*\langle v|w\rangle$ but in Wikipedia and other school's lecture notes they say $\langle v|\alpha w\rangle=\alpha^*\langle v|w\rangle$. $\alpha^*$ denotes the complex conjugate of the constant $\alpha$. So I was wondering which one is more natural? Or is it ok as long as I keep it consistent?
This is a difference between the physicist notation, which is linear in the second term, and the mathematical notation, which is linear in the first term. The physicist notation is better on account of lending power to Dirac notation. However, there is no real conflict, as long as you know your author's convention.