Geometric Properties of Banach Space and Manifold

The subject of this book, geometric properties of manifolds, is a simplified and well organized for students of graduate program and above in applied functional analysis theory. The main discussion of this book relies on smoothness, boundedness, duality, and Gateaux differentiability of manifolds. It consists of two parts. The first part discusses the geometric properties of Banach spaces, such as strict convexity, uniform convexity, smoothness, uniform smoothness, reflexivity and the Kadec-Klee property, duality map which will play crucial roles in the study of iterative algorithms for nonlinear operators in various Banach spaces. Also we define a real function called the modulus of convexity and the modulus of smoothness depending on the Banach space under consideration. The second part of the book reflects the basic theories on manifolds and their geometric properties. Thus, properties are very widely not applicable in studies on modern functional analysis. The book solves some basic problems on surface of a manifold, such as problems on tangent spaces, isometry on smooth Riemannian manifold, local charts, and Levi-Civita connection.