MATHEMATICAL ANALYSIS Volume-II

This is a multipurpose text, it is a graduate course covering differentiation on normed spaces and integration with respect to complex and vector-valued measures. One also has the option of limiting all to En, or taking Riemannintegration before Lebesgue theory (we call it the “limited approach”). The proofs and definitions are so chosen that they are as simple in the general case
as in the more special cases. In a nutshell, the basic ideas of measure theory
are given in Chapter 7. In Chapter 6 (Differentiation), we have endeavored to present a modern theory, without losing contact with the classical terminology and notation. (Otherwise, the student is unable to read classical texts after having been
taught the “elegant” modern theory.) This is why we prefer to define derivatives
as in classical analysis, i.e., as numbers or vectors, not as linear mappings. The
latter are used to define a modern version of differentials. In Chapter 9, we single out those calculus topics (e.g., improper integrals) that are best treated in the context of Lebesgue theory. Our principle is to keep the exposition more general whenever the general case can be handled as simply as the special ones (the degree of the desired specialization is left to the instructor). Often this even simplifies matters—for example, by considering normed spaces instead of En only, one avoids cumbersome coordinate techniques. Doing so also makes the text more flexible.

Mathematical Analysis, Volume 1 is also available for separate purchase.