Is there any difference between $\sum_{ij}$ and $\sum_i\sum_j\;?$ I was reading Tensors by Feynman, where he said:

[...]$$u_P=\tfrac{1}{2}\sum_i\sum_j\alpha_{ij}E_iE_j\;.\tag{31.7}$$
The energy density $u_P$ is a number independent of the choice of axes, so it is a scalar. A tensor has then the property that when it is summed over one index (with a vector), it gives a new vector; and when it is summed over both indexes (with two vectors), it gives a scalar.

Got the point what Feynman wanted to say.
Then he discussed about the tensor of inertia and wrote the kinetic energy as:

$$\text{KE}=\tfrac{1}{2}\sum_{ij}I_{ij}\omega_i\omega_j.\tag{31.17}$$

Now, we know $\text{KE}$ is scalar; so it must be 'summed over both indexes (with two vectors)' as said by Feynman.
However, the notation of summation is not the same he used in $(31.7)$ as in $(31.17);$ in the former, the notation is $\sum_i\sum_j$ while in the later the notation is $\sum_{ij}\;.$ Since both are scalar and the summation is computed over two vectors in both the equations, can I infer from Feynman's statement 'when it is summed over both indexes (with two vectors), it gives a scalar' that $\sum_{ij}$ and $\sum_i\sum_j$ are necessarily the same?
 A: The way I interpret your question:

If $a_{ij}$ are scalars indexed by some subset of $\Bbb N^2$, then are the two summation signs the same thing?

I'll assume you know some basics of sequence and series theory. (If not, then this answer is not helpful to you.)
In the absence of absolute convergence, $\sum_{ij}$ itself makes little sense because it's ambiguous as to the order in which to carry out the summation. And check out this famous theorem which says that if you have only conditional convergence, the order does matter. However, when we indeed have a.c., this problem is automatically resolved since the order doesn't matter anyway. So, if the author has explicitly written $\sum_{ij}$ in the first place, and if he's mathematically literate (which is of course true in your case), then he's already (maybe implicitly) assumed absolute convergence, so the summation order doesn't matter and the two signs are essentially saying the same thing. 
A: $\sum_{ij}$ is just a shorthand for $\sum_i\sum_j$. They both mean "sum over index $i$ and index $j$".
In some cases the order counts, in that case the notation with two sums is preferable.
