This book is the outgrowth of the lectures delivered on functional analysis and allied topics to the postgraduate classes in the Department of Applied Mathematics, Calcutta University, India. I feel I owe an explanation as to why I should write a new book, when a large number of books on functional analysis at the elementary level are available. Behind every abstract thought there is a concrete structure. I have tried to unveil the motivation behind every important development of the subject matter. I have endeavoured to make the presentation lucid and simple so that the learner can read without outside help.

The first chapter, entitled ‘Preliminaries’, contains discussions on topics of which knowledge will be necessary for reading the later chapters. The first concepts introduced are those of a set, the cardinal number, the different operations on a set and a partially ordered set respectively. Important notions like Zorn's lemma, Zermelo's axiom of choice are stated next. The concepts of a function and mappings of different types are introduced and exhibited with examples. Next comes the notion of a linear space and examples of different types of linear spaces. The definition of subspace and the notion of linear dependence or independence of members of a subspace are introduced. Ideas of partition of a space as a direct sum of subspaces and quotient space are explained. ‘Metric space’ as an abstraction of real line ℝ is introduced.