Inner Product Spaces

To formalize the comments as an answer:

The difference between requiring $$(\alpha u,v)=\alpha(u,v)\quad\text{ (mathematician's definition)}$$ and $$\langle u, \alpha v\rangle=\alpha\langle u,v\rangle\qquad\quad\,\,\text{ (physicist's definition)}$$ is purely one of convention, and the two definitions are equivalent as $(u,v)=\langle v,u\rangle$. There's no intrinsic reason to choose either, though if you work exclusively with one for long enough, you might come to regard the other as an abomination. In general, it is always advisable to keep an eye to which convention is being used.

The physicist's definition does have the advantage that it extends well to Dirac notation, in the sense that matrix elements such as $\langle \phi|\hat{A}|\psi\rangle$ are linear in $\psi$, so that the state $\hat{A}|\psi\rangle$ corresponds to the operator-acting-on-a-vector notation $Av$. If the bracket were linear in $\phi$ then we'd have to make operators act to their left. This is again an OK convention but no one uses it.