Differentiation of Operators

This chapter is essentially a brief introduction to non-linear functional analysis. First, we define the Gâteaux and Fréchet derivatives of generally non-linear operators between linear vector spaces and we investigate their properties in some considerable detail. We prefer to use the term “derivative” although the term “differential” provides perhaps a better description of these notions. We shall see that the Fréchet derivative will be particularly useful in designing a linear approximation to a non-linear operator. The structural properties of a Fréchet derivative enables us to introduce higher order Fréchet derivatives. We shall then attempt to extend the concept of the classical Riemann integral of numerical functions, so that it can be applicable to the integration of operators. We shall examine several important properties of this integral and we shall eventually manage to obtain some kind of a Taylor’s expansion for operators. We shall then discuss the Newton method, which is a very effective tool in determining the solutions of general operator equations by successive approximations, together with certain criteria for convergence and the method of steepest descent (or ascent) which gives rise to quite a useful algorithm to find extrema of real-valued non-linear functional. We finally conclude this chapter by presenting quite a general form of the famous implicit function theorem of classical analysis.