Chapter K Differential Calculus In the second half of this book, starting from Chapter F, we have worked on developing a thorough understanding of function spaces, may it be from a geometric or analytic viewpoint. This work allows us to move towards a variety of directions. | Chapter K Differential Calculus In the second half of this book starting from Chapter F we have worked on developing a thorough understanding of function spaces may it be from a geometric or analytic viewpoint. This work allows us to move towards a variety of directions. In particular we can now extend the classical differential calculus methods to the realm of maps defined on suitable function spaces or more generally on normed linear spaces. In turn this generalized calculus can be used to develop a theory of optimization in which the choice objects need not be n-vectors but members of an arbitrary normed linear space as in calculus of variations control theory and or dynamic programming . This task is carried out in the present chapter. We begin with a quick retake on the notion of derivative of a real-to-real function and point to the fact that there are advantages of viewing this notion as a particular linear functional as opposed to a number. Once this point is understood it becomes straightforward to extend the notion of derivative to the context of functions whose domains and codomains lie within arbitrary normed linear spaces. Moreover the resulting derivative concept called the Fréchet derivative inherits many properties of the derivative that you are duly familiar with from classical calculus. We study this concept in fair detail here go through several examples and extend some well-known results of calculus to this realm such as the Chain Rule the Mean Value Theorem etc. Keeping an eye on optimization theoretic applications we also revisit the theory of concave functions this time making use of Fréchet The use of this work is demonstrated by means of a brief introduction to infinite dimensional optimization theory. Here we see how one can easily extend first- and second-order conditions for local extremum of real functions on the line to the broader context of real maps on normed linear spaces. We also show how useful concave functions are
