Inner Product Spaces

Question

I am trying to reconcile the definition of Inner Product Spaces that I encountered in Mathematics with the one I recently came across in Physics. In particular, if $(,)$ denotes an inner product in the vector space $V$ over $F$:

1. $(u + v, w) = (u, w) + (v, w) \text{ for all } u, v, w \in V$,

2. $(\alpha v, w) = \alpha(v, w) \text{ for all } v, w \in V$ and $\alpha \in F$,

3. $(v, w) = (w, v)^* \text{ for all } v, w \in V$, (* denotes complex conjugation)

were some of the properties listed in my mathematics course.

In physics, however, the inner product was said to be linear in the second argument and $(v,\sum(\lambda_i w_i)) = \sum(\lambda_i (v,w_i))$ where $v$ and $w_i$ are kets in Hilbert space and $\lambda_i$ are complex numbers.

To me, these properties of an Inner Product are not compatible. If the first definition of inner product is correct, then I think $(v,\sum(\lambda_i w_i)) = \sum((\lambda_i)^* (v,w_i))$ where $^*$ denotes complex conjugation.

Answer 1

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.

Answer 2

The inner product used in quantum mechanics is sesquilinear, as opposed to just linear. A good reference to read up on this is Hassani: Mathematical Physics.