I'm trying to understand the definition of maps between vector spaces (in normed vector spaces) listed in the following link: http://ocw.mit.edu/courses/mathemat...anifolds-fall-2004/lecture-notes/lecture3.pdf On the surface, this seems similar to what I expected from the definition of a derivative of a function R -> R. But when I looked at the definition for the small o notation, I got confused because I don't understand how the kind of continuity described implies that the LHS divided by ||x - x'|| goes to 0 as x approaches x'. Is this the same kind of o notation that is used when you talk about things like time complexity of algorithms? Even if it is, it seems kind of strange because you usually use this notation when the function argument gets very large. On the other hand, p. 15 - 17 of http://www.math.uiuc.edu/~tyson/595chapter3.pdf seem to describe exactly what I was expecting from a generalization of the derivative for a map between vector spaces where it makes sense. I think the definition here should probably be equivalent to the link I posted above, but I think I'm probably missing something. Are the two definitions here equivalent and am I missing something about the o notation and whether it implies things grow more slowly? Are they even actual direct generalizations of the R -> R case?