I have seen an operator $A$ called the Infinitesimal Generator.
Given $b: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\sigma: \mathbb{R}^n \rightarrow \mathbb{R}^{n m}$ and $f:\mathbb{R} \rightarrow \mathbb{R}^n$,
$Af(x) = \sum_i b_i \frac{\partial f}{\partial x_i}(x) + \frac{1}{2} \sum_{i,j}(\sigma(x)\sigma(x)^T)_{i,j}\frac{\partial^2f}{\partial x_i \partial x_j} (x)$
My question is about the subscripts from $\sum_{i,j}$ . Am I to interpret such a sum to mean the same thing as $\sum_i \sum_j$? If so, then in applications of this operator, I should expect not to see the $\frac{1}{2}$ coefficient of the term $\frac{1}{2} \sum_{i,j}(\sigma(x)\sigma(x)^T)_{i,j}\frac{\partial^2f}{\partial x_i \partial x_j} (x)$, since each inner term should appear twice by symmetry (swapping i with j).
However, I see an example application which says:
$A f(t,x) = \frac{\partial f}{\partial t} (t,x) + \frac{1}{2} \frac{\partial ^2 f}{\partial x ^2} (t,x)$
when $n,m = 2$, $b(t,x) = (0,1)$ and $\sigma_{i,j} = 1$ for $i,j=2$ and $\sigma_{i,j} = 0 $ else.
This notation was encountered at http://en.wikipedia.org/wiki/Infinitesimal_generator_(stochastic_processes)#Definition
Each entry-wise distinct pair appears twice. So there are two terms such as $(i,j)=(1,2)$ and $(i,j)=(2,1)$ but one term of $(i,j)=(1,1)$. So the $1/2$ term remains on all pairs where $i=j$.
As an analogy consider expanding $(\sum_i x_i)^2$, which has a factor of 1 infront of each $x_i^2$ term but a factor of 2 for $x_ix_j$.