Differentiation in Banach Spaces

Zigang Pan

Chapter Chapter 9 in Measure-Theoretic Calculus in Abstract Spaces, 2023, pp 257-334 from Springer

Abstract: Abstract In a Banach space, all the necessary ingredients are present for the study of calculus on the space. This topic of differentiation on Banach space is relatively well understood and has significant body of presentation in Luenberger (Optimization by Vector Space Methods. Wiley, New York, 1969). I follow closely the results on Fréchet derivative for functions between Banach spaces. I take extra care in the results for functions with general domains (not simply open domains). There, an extra condition is needed for the local shape of the domain at the point of differentiation. For high order derivatives (including partial derivatives), we inevitably deal with tensor operators. The chain rule (Theorem 9.18) and mean value theorems are established. I introduce the assumptions under which the kth order derivative f (k)(x 0) is a k-fold symmetric operator. Interchange order of differentiation can be carried out according to Propositions 9.31 and 9.32. The Taylor’s Theorem 9.48 is presented. The usefulness of the differentiation in Banach space is because of various mapping theorems (everything you can think off) in Sect. 9.5 and analytic functions in Sect. 9.9. In Sect. 9.7, we present sufficient conditions under which the limit operation and differentiation operation in series can be interchanged. In Sect. 9.8, we give the definition and properties of basic tensor operations. The chapter ends with the section on Newton’s Method.

Date: 2023
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DOI: 10.1007/978-3-031-21912-2_9

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