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This book presents Advanced Calculus from a geometric point of view: instead of dealing with partial derivatives of functions of several variables, the derivative of the function is treated as a linear transformation between normed linear spaces.

Aims to convey 3 principal developments in the evolution of information theory, including Shannon's interpretation of Boltzmann entropy as a measure of information yielded by an elementary statistical experiment and basic coding theorems on storing messages and transmitting them through noisy communication channels in an optimal manner.

It gives Liouville's Theorem on the existence of invariant measure, entropy theory leading up to Kolmogorov-Sinai Theorem, and the topological dynamics proof of van der Waerden's theorem on arithmetical progressions.

Provides an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups.

The material presented in this book is suited for a first course in Functional Analysis which can be followed by Masters students. The book includes a chapter on compact operators and the spectral theory for compact self-adjoint operators on a Hilbert space.