Invisible wrote:
> 
> For example, the Fourier transform allows you to construct any function 
> from sine and cosine functions.
> 

This isn't quite true over the reals, even assuming you're only looking 
for functions with a given period.  For example the function which is 
zero everywhere except being 1 at a single point will generate the same 
Fourier representation as the constant zero function since it will have 
the same integrals.

I think (no proof) that you can reconstruct any function up to the 
addition of a function which is nonzero over an area of zero `volume' 
though (assuming you don't count things like a delta functions).  Not 
that it matters for what you're doing of course, but you seem like the 
sort of chap who might find it interesting.

> 
> So, like, how do you tell if two functions are orthogonal? And how do 
> you tell when a set of them is complete?

What are you using these function for?  There may be better or worse 
ways to do things depending on what you want.

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