Differentiating maps between vector spaces

[modnarandom]

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?

 

[RUber]

The little o is kind of the opposite of the big O notation. Where the big O means "on the order of" ... or $F(n) = \mathcal{O}(n^2) \implies \exists C: F(n) \approx Cn^2$,
The little o can be defined as $F(n) = o(n) \implies \lim F(n)/n = 0$ or F(n) is dominated by n in the limit.

 

[modnarandom]

Oops! I forgot to mention that I initially interpreted o(||x - x'||) to be the kind of little o you mentioned. It's just that I don't know how it makes sense in the context of the definition of a derivative of a map between vector spaces.

 

[RUber]

It seems to me that the two sources are saying the same thing.
$\| f(x) - f(x') - L(x-x') \| = o(\|x-x'\|)$ means $\lim \frac{\| f(x) - f(x') - L(x-x') \|}{\|x-x'\|} = 0$.

 

[modnarandom]

Ok, thanks! But doesn't your first post assume that you're considering the limit as n goes to infinity? Or is the definition something else? I guess I'm not sure how this definition follows from o being a function continuous at 0.

 

[RUber]

I suppose they are adjusting the definition to describe the behavior in the limit as x goes to x'. I agree that normally the little-o and big-O notations are used for large arguments, so perhaps it is implicit that there is a series x_n such that x_n goes to x as n goes to infinity or some business like that.