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This book bridges a gap between introductory and high-level textbooks, showing how calculus on normed vector spaces extends the basic calculus of functions of several variables. Includes many non-trivial applications of the theory and interesting exercises.

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This book serves as an introduction to calculus on normed vector spaces at a higher undergraduate or beginning graduate level. The prerequisites include basic calculus and linear algebra, as well as a certain mathematical maturity. All the important topology and functional analysis topics are introduced where necessary.

In its attempt to show how calculus on normed vector spaces extends the basic calculus of functions of several variables, this book is one of the few textbooks to bridge the gap between the available elementary texts and high level texts. The inclusion of many non-trivial applications of the theory and interesting exercises provides motivation for the reader.

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Preface.- 1 Normed Vector Spaces.- 2 Differentiation.- 3 Mean value theorems.- 4 Higher derivatives and differentials.- 5 Taylor theorems and applications.- 6 Hilbert spaces.- 7 Convex functions.- 8 The inverse and implicit mapping theorems.- 9 Vector fields.- 10 The flow of a vector field.- 11 The calculus of variations: an introduction.- Bibliography.- Index

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From the reviews:

"Coleman (Laboratoire Jean Kuntzmann, France) generalizes the standard theorems of this subject to an arbitrary, hence possibly infinite, dimensional normed vector space, including the special cases of Banach spaces and Hilbert spaces. ... Best suited for advanced undergraduate or graduate students, the text delivers the level of rigor and mathematical sophistication that one in the discipline expects from a volume published by Springer. Summing Up: Recommended. Upper-division undergraduates, graduate students, and researchers/faculty." (D. S. Larson, Choice, Vol. 50 (10), June, 2013)
"It covers all topics of classical calculus in the broad setting of real normed spaces. ... The style used in giving a complete overview of all these topics of differential calculus is a really nice one, introducing concepts whenever they are needed ... . Each chapter contains a variety of interesting exercises which help to complement the material being studied. Therefore, to finish, I highly recommend this book to teachers and researchers on the subject." (Jesús Ferrer, Mathematical Reviews, March, 2013)

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