Differential Calculus on Banach Spaces and Extrema of Functions

Abstract

As is well known for functions on Euclidean spaces, the local behavior is determined by the existence of derivatives in a point and the properties of these derivatives. We are going to show that this holds for functions on infinite-dimensional Banach spaces too. Accordingly we begin by introducing the Fréchet derivative and study its basic properties. This allows us to prove the Taylor expansion for functions which are defined on open subsets of a Banach space and which have values in a Banach space in substantial generality. Equipped with this Taylor expansion local extrema of functions can be characterized through properties of the first few Fréchet derivatives (necessary respectively sufficient conditions of Euler–Lagrange). Next we show that convexity of a function can be characterized through monotonicity of its Fréchet derivative. This allows us to show that convex functions are weakly lower semicontinuous and that every critical point of a convex function is actually a local minimum. We conclude by introducing two more concepts of a derivative, namely the directional derivative of Gâteaux and the concept of variations. In practice the calculation of the Gâteaux derivative is much easier than that of the Fréchet derivative. We present sufficient conditions under which Gâteaux differentiability implies Fréchet differentiability.