Smooth Analysis in Banach Spaces

This book is about the subject of higher smoothness in separable real Banach spaces. It brings together several angles of view on polynomials, both in finite and infinite setting. Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available?

The subject of infinite dimensional real higher smoothness is treated here for the first time in full detail, therefore this book may also serve as a reference book.

• Detailed exposition of the foundations of the subject
• Connects several areas of research, namely polynomials on finite and infinite dimensional spaces, variational principles, renormings and structure of Banach spaces

Petr Hájek, Academy of Sciences of the Czech Republic, Prague, Czech Republic; Michal Johanis, Charles University, Prague, Czech Republic.

"The authors have presented an impressive piece of work […]. However technical the material is they have provided detailed proofs and illuminating comments. To make their monograph self-contained they have also incorporated a lot of background material, often with full proofs." Zentralblatt für Mathematik