I have a very very quick question about some notation used in Spivak's Calculus on Manifolds.
At one point in the book he states that any $k$-form may be written as $$ \omega = \sum _{i_{1} < \,\cdots \, < i_{k}}\omega _{_{i_{1} ,\, \dots \, , \, i_{k}}} \, dx^{i_{1}} \wedge \cdots\wedge dx^{i_{k}} $$ where $dx^{i}$ are the elements of the dual basis. Now, I understand why is it that the dual basis can be used to form the basis for alternating $k$-tensors, and also why there are exactly $$ \frac{n!}{k!\, (n-k)!} $$ elements in said base (if they are defined over some vector space $V$ of dimension $n$). What I don't get is what this means $$ \sum _{i_{1} < \,\cdots \, < i_{k}} $$ I've never seen quite anything like this before. What are you summing over? How does that translate into the summands? I've got no clue, and Spivak sure doesn't try to make it clear. Any help would be appreciated :) If you can show a simple example that'd be great, too.
In statistics a sum of a similar form may have appeared when studying covariance. To make it simple, consider the sum $\sum_{1 \leq i < j \leq n}a_{i}a_{j}$. You can fix every $i$ and let $j$ run to get the sum, which is $=$ $$ (a_{1}a_{2} + a_{1}a_{3} + \cdots + a_{1}a_{n}) + (a_{2}a_{3} + a_{2}a_{4}+\cdots +a_{2}a_{n}) + \cdots + (a_{n-1}a_{n}). $$