Abstract
The purpose of this book is to lay down the foundations for the abstract theory of C^k-smoothness in infinite-dimensional real Banach spaces and investigate its intimate connections with the structural properties of the underlying spaces. The classical Banach spaces c0 and lp play a decisive role throughout the subject.
Indeed, the supply and properties of smooth functions on a given Banach space depend heavily on its linear structure. The development of the theory relies on methods and results from the local theory of Banach spaces, tensor products, several topological tools, and of course the structural theory of Banach spaces and general analysis.
A significant portion of the book deals with various aspects of the theory of polynomials, as polynomials are tightly connected with the higher smoothness via the Taylor and the Converse Taylor theorems. A large part of the theory is also devoted to the study of approximations of continuous mappings by means of mappings with prescribed smoothness properties.
The subject of infinite dimensional real higher smoothness, including real analyticity, is treated here for the first time in full detail. The book contains a large number of very recent or completely new results with streamlined proofs.
The book will be of interest to both researchers and graduate students in analysis, and may also serve as a reference book.