Analysis in Vector Spaces

This text is intended for a second course in analysis, concentrating on differentiation and integration of functions of several variables. The material is presented as a unified course, rather than as a set of unrelated and slightly related topics, and the underlying and unifying concepts is vector spaces. The book successfully provides a firm mathematical foundation for further study and established an indispensable background for further studies in mathematics or related disciplines that involve heavy use of mathematical techniques and concepts. The books also provides exposure to a variety of other mathematical topics such as basic point-set topology, measure theory, differential geometry, and the theory of manifolds. Basic notations concerning sets, functions, real numbers, and the like are all carefully presented in the background chapters. All statements made by the authors are rigorously proved with explicit reference to earlier results, thus eliminating the need to search through earlier portions of the text to find out which results are being applied and where they were first proved. The authors retain the connection to the rich historical background of the subject, and they ensure that the underlying intuitive ideas play an essential part in the presentation of the proofs.