Conflicting Conventions

This has been irking me ever since I cracked open Griffiths. Why do mathematics and physics have the opposite conventions when it comes to inner products?

By definition, an inner¹ product must satisfy the following for all vectors a, b, and c, scalar λ, in the appropriate vector space V:

• conjugate symmetry:
• linearity in the first variable:
• positive-definite-ness: and iff
  

For example, the standard inner product for the space (over the field of complex numbers, of course) is , where a_i and b_i denote the ith component of vectors a and b, respectively. A really cool, and useful, inner product for the space (the set of functions that are continuous on [-1, 1]) is (well, assuming that the functions have only one variable and all that). The interval can be tweaked as appropriate (e.g. integrating from to if the functions allow that).

None of this is new, I hope. My professor defines inner products this way. My linear algebra textbook defines inner products this way. Wikipedia defines inner products this way.

Why is it that physicists insist on defining them backwards? Linearity in the second variable instead of the first, and all that. Worse yet, they have a different set of notation! Instead of saying , they say . And wait until we get to things like notation for self-adjoint operators.

The only good thing about the physicist convention is that looks nicer than . And that won't be a problem in the first place if one uses the bar to denote "complex conjugate".

I am looking at you, quantum mechanics. Just do it the mathematicians' way, will you²? This is aggravating: what's a mathie-chemist student supposed to do?

¹I might end up calling an inner product a "dot product" by accident. Strictly speaking, they aren't the same thing (the dot product refers to the standard inner product over the set of reals). Sorry.
^{2 3}Not that anyone will listen. Conventions are hard to change.
³I am using superscripts instead of asterisks so that I don't overload the asterisk symbol too much.