Even and odd functions don't form a partition of the space of all functions, but they do span it as a vector space. What's more, the vector space of functions i

Even and odd functions don't form a partition of the space of all functions, but they do span it as a vector space. What's more, the vector space of functions is the direct sum of the even and odd functions: every function has a unique decomposition as a sum of an even and an odd function. So the problem there is that vector spaces don't behave the same way as sets.