TITLE: Cross Product for functions
QUESTION [6 upvotes]: So functions are just uncountabley-infinite dimensional vectors, and as such there's a nice generalization of the inner product between two functions (the integral of their product).  Is their a similar generalization for the cross product between two functions?!

REPLY [2 votes]: Thinking of functions as uncountably-infinite vectors may lead you wildly astray, so I'd be careful of that.
The cross product is a fairly unique binary operation, as it turns out.  For any vector space $V$, we have an inner product $\langle\cdot,\cdot\rangle:V\times V\to \mathbb R$ (we're just going to assume these are real vector spaces). However, in most circumstances we DO NOT have an operation from $V\times V\to V$.  The notable exceptions are the cross product in $\mathbb R^3$ and $\mathbb R^7$ (see here).
You can lift the idea of an inner product to functions, though. Have you had any functional analysis? If you haven't this idea may be a little hard to tackle.