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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.
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