1.1 Differential Calculus

Let us begin with an outline, without proofs, of differential calculus in Banach spaces. For proofs and more details we refer to [20], Chapters 1 and 2.

The Fr ch t derivative. Let X, Y be Banach spaces and let L ( X, Y ) denote the space of linear continuous maps from X to Y. For A ? L ( X, Y ) we will often write Ax or A[ x] instead of A ( x ). Endowed with the norm

L ( X, Y ) is a Banach space. If U ? X is an open set, C(U, Y) denotes the space of continuous maps f : U ? Y.

Definition 1.1

We say that f : U Y is (Fr ch t) differentiable at u ? U with derivative d f ( u ) ? L ( X, Y ) if

f is said differentiable on U if it is differentiable at every point u ? U.

From the definition it follows that if f is differentiable at u ? U then f is continuous at u.

In order to find the derivative of a map f one can evaluate, for all h ? X, the limit

If A _u ? L (