The derivative is a linear transformation. This may not be entirely clear from a beginning calculus course because they like to say it is a slope which is a number. As observed by Deudonne, “...In the classical teaching of Calculus, this idea (that the derivative is a linear transformation) is immediately obscured by the accidental fact that, on a onedimensional vector space, there is a onetoone correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a number instead of a linear form. This slavish subservience to the shibboleth^{1} of numerical interpretation at any cost becomes much worse when dealing with functions of several variables...” He is absolutely right.
The concept of derivative generalizes right away to functions of many variables but only if you regard a number which is identified as the derivative in single variable calculus as a linear transformation on ℝ. However, no attempt will be made to consider derivatives from one side or another. This is because when you consider functions of many variables, there isn’t a well defined side. However, it is certainly the case that there are more general notions which include such things. I will present a fairly general notion of the derivative of a function which is defined on a normed vector space which has values in a normed vector space. The case of most interest is that of a function which maps an open set in F^{n} to F^{m} but it is no more trouble to consider the extra generality and it is sometimes useful to have this extra generality because sometimes you want to consider functions defined, for example on subspaces of F^{n}and it is nice to not have to trouble with ad hoc considerations. Also, you might want to consider F^{n} with some norm other than the usual one.
For most of what follows, it is not important for the vector spaces to be finite dimensional provided you make the following definition which is automatic in finite dimensions.

To save notation, I will use
Let U be an open set in X, and let f : U → Y be a function.
Definition 17.1.2 A function g is o
 (17.1) 
A function f : U → Y is differentiable at x ∈ U if there exists a linear transformation L ∈ℒ

This linear transformation L is the definition of Df
Note that from Theorem 11.5.4 the question whether a given function is differentiable is independent of the norm used on the finite dimensional vector space. That is, a function is differentiable with one norm if and only if it is differentiable with another norm. In infinite dimensions, this is not clearly so and in this case, simply regard the norm as part of the definition of the normed linear space which incidentally will also typically be assumed to be a complete normed linear space.
The definition 17.1 means the error,

converges to 0 faster than
 (17.2) 
or equivalently,
 (17.3) 
The symbol, o

and other similar observations hold.
Proof: First note that for a fixed vector, v, o

Now suppose both L_{1} and L_{2} work in the above definition. Then let v be any vector and let t be a real scalar which is chosen small enough that tv + x ∈ U. Then

Therefore, subtracting these two yields
Lemma 17.1.4 Let f be differentiable at x. Then f is continuous at x and in fact, there exists K > 0 such that whenever

Also if f is differentiable at x, then

Proof: From the definition of the derivative,

Let
The last assertion is implied by the first as follows. Define

Then lim_{}

This establishes the second claim. ■
Here