Let X,Y be

normed spaces with

norms || ||

_{X} and || ||

_{Y}.

A

function f: X -> Y is called

**differentiable** in the point x of X, iff there is a

continuous linear map A and the

limit
f(x+h) - A(h) - f(x)
lim ---------------------------
||h|| -> 0 ||h||

exists and equals 0 (||h||=||h||

_{X}, h of X).

A is called the

**derivative** of f in x and is usually written as Df(x). (This is a linear map, to get any values you would have to write Df(x)(z).)

Let L(X,Y) the normed space of continuous linear functions from X to Y.

If the derivative of f exists in an open neighborhood of x, then the derivative of the map Df: X -> L(X,Y), x |-> Df(x) might exist. It's called the **second derivative** of f in x, written as D^{2}f(x).

If the second derivative exists in an open neighborhood of x, then the derivative of D^{2}f : X -> L(X,L(X,Y)) might exist and is called the **third derivative**.

In fact the **n-th derivative** is the derivative of the function D^{n-1}f : X -> L(X,...L(X,Y))...)), x-> D^{n-1}f(x).

However these "stacked" spaces of linear functions L(X,...L(X,Y))..)) are difficult to use. Therefore one uses the fact that L(X,..L(X,Y)...) with n L's stacked is isometric to B(X,Y,n) is the space of n-linear continuous functions (Note: B(X,Y,n) is not canonical for this space, I just made it up)

The isomorphism is defined per: h of L(X,...L(X,Y)..) goes to g of B(X,Y,n) via g(x_{1},...,x_{n}):= h(x_{1})...(x_{n}).

So one takes as the n-th derivative the function D^{n}f(x_{1})...(x_{n}) instead of
D^{n}f(x_{1}) of L(X,...L(X,Y)..) (n-1 L's stacked)

The function f is called n times **continuous differentiable** in x iff the map D^{n}f: X -> B(X,Y,n), x |-> D^{n}f(x) is continuous (Note: this is not a linear map !)

Now comes the question: "What has this to do with the usual derivative of **R**^{1} -> **R**^{1} ?"

The derivative of **R**^{1} is scalar. Multiplication with a scalar is the form of linear maps from **R**^{1} to **R**^{1}. Set Df(x)(y) := f'(x)· y and you get the above form.

This definition allows you to differentiate in really sick spaces like function space, spaces of matrices etc.

The derivatives are quite difficult to determine there but some simple laws still hold:

- The derivative of a continuous linear map is the map itself at any point of X.
- The chain rule always holds: D(f(g))(x)(y) = (Df)(g(x))(Dg(x)(y)) where (Df)(g(x)) is the derivative of f at point g(x), the formula means: to get the image of y under D(f(g)) first apply Dg(x) on y and then apply (Df)(g(x)) ( f(g) is the function you get when you apply f to the images of g)