ABSTRACT
As seen in Chapters 1 and 2, functional analysis involves abstraction and formalization of ideas studied in physics and classical mathematics. It seeks to identify the most primitive features of elementary analysis, geometry, and calculus to give them order and structure, and to establish their interrelationship. It simultaneously unifies a bunch of ideas and extends them to areas which have been explored in the framework of classical mathematics or different branches of science and technology. A Hilbert space is a special type of Banach space in which the underlying vector space is equipped with a structure called an inner product or a scalar product which provides the generalization of geometrical concepts such as direction, orthogonality (perpendicularity of two vectors), and angle between two vectors. The inner product is nothing but a generalization of the dot product or scalar product of vector calculus. Hilbert space with a rich structure of inner product is a powerful apparatus to tackle problems of diverse fields of classical mathematics like linear equations, variational methods, approximation theory, differential equations. Besides this, Hilbert space methods have played a vital role in resolving problems of numerous fields of science and technology. A few recent applications will be discussed in Chapters 10 and 11.
