# Differentiating maps between vector spaces

• modnarandom
In summary: I don't understand how the big-O notation works. It looks like it might imply that something grows more slowly, but I'm not sure what it means.
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?

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.

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.

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 ##.

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.

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.

## 1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, known as vectors, and a set of operations that can be performed on these vectors. These operations include addition and scalar multiplication, and they must follow specific rules and properties to qualify as a vector space.

## 2. How do you differentiate maps between vector spaces?

Differentiating maps between vector spaces involves finding the derivative of a map, which is a function that assigns an output vector to each input vector. This can be done by applying the chain rule and product rule, as well as other rules of differentiation, to the components of the map.

## 3. What is the purpose of differentiating maps between vector spaces?

The purpose of differentiating maps between vector spaces is to understand how small changes in the input vectors affect the output vectors. This is useful in many areas of mathematics and science, including physics, engineering, and economics.

## 4. Can differentiating maps between vector spaces be applied to real-life problems?

Yes, differentiating maps between vector spaces has numerous real-life applications. For example, it can be used to optimize processes in engineering, analyze trends in economics, and model physical systems in physics.

## 5. Are there any limitations to differentiating maps between vector spaces?

One limitation of differentiating maps between vector spaces is that it can only be applied to maps that are differentiable, meaning they have a well-defined derivative at every point in their domain. Additionally, the dimensionality of the vector spaces involved can also impact the complexity of the differentiation process.

• Differential Geometry
Replies
21
Views
1K
• Differential Geometry
Replies
4
Views
2K
• Differential Geometry
Replies
10
Views
1K
• Differential Geometry
Replies
20
Views
2K
• Differential Geometry
Replies
3
Views
512
• Differential Geometry
Replies
11
Views
3K
• Differential Geometry
Replies
12
Views
3K
• Differential Geometry
Replies
7
Views
2K
• Differential Geometry
Replies
4
Views
2K
• Differential Geometry
Replies
4
Views
2K