This is a foundational textbook designed to introduce students to the core ideas and techniques of linear analysis with clarity and mathematical rigour. The text bridges the gap between elementary linear algebra and advanced functional analysis, making it suitable for advanced undergraduates and beginning graduates in mathematics, physics and engineering. The book is designed to serve as one-semester course text in linear functional analysis. It will also be valuable to research scholars, faculties of university, colleges and various engineering institutes as well as professionals who wish to acquire a solid foundation in functional analysis and its applications. It will help the readers, familiar with finite-dimensional concepts to move towards rich and subtle theory of normed and inner product spaces, bounded linear operators and fundamental theorems that govern them. The concepts of normed linear spaces and inner product spaces are discussed emphasising geometric intuition alongside formal proofs. Key topics such as Banach and Hilbert spaces are explained, followed by a systematic study of bounded linear operators, dual spaces and continuous linear functionals. Important theorems like Hahn-Banach theorem, the open mapping theorem, the uniform boundedness principle and the closed graph theorem are presented with detailed proofs and illustrative examples. The theory is reinforced by several examples and exercises. All exercises have been solved at the end of the book. Special attention is given to operators on Hilbert spaces including self-adjoint, unitary and normal operators as well as projections and spectral ideas. The book also serves as a solid foundation for further study and research in functional analysis, operator theory, harmonic analysis and related fields.
About the Author :
Dr. Ajay Kumar is a distinguished mathematician currently serving as a senior scientist and formerly a professor in the Department of Mathematics along with serving at various positions like Head of the Department, Dean of Faculty of Mathematical Sciences and Dean Research at the University of Delhi. He has been teaching functional analysis and related analysis papers for more than 35 years. His research spans several core areas of modern analysis including harmonic analysis, representations of locally compact groups, potential theory on stratified and nilpotent Lie groups, advanced functional analysis, C*-algebras, operator spaces and systems, and complex analytic methods in partial differential equations.
Over his career, Dr. Kumar has published more than 90 research papers in premier international journals, such as Transactions of the American Mathematical Society, Journal of Functional Analysis, Potential Analysis, Journal of Geometric Analysis, Pacific Journal of Mathematics, Mathematische Zeitschrift, etc. His work reflects collaborations with renowned mathematicians across several countries. He has supervised 24 Ph.D. theses and 19 M.Phil. dissertations, contributing significantly to the development of emerging scholars in mathematics. He is a fellow of the National Academy of Sciences, India (NASI), and recipient of NASI Senior Scientist Platinum Jubilee Fellowship. He has been awarded over a dozen prestigious international research fellowships, including DAAD (Germany), Commonwealth Staff (UK), CIES (France), Royal Society (London), JSPS (Japan), and DFG (Germany).
Ashish Bansal
Dr. Ashish Bansal is an accomplished mathematician and educator serving as a professor in the Department of Mathematics at Keshav Mahavidyalaya, University of Delhi. His doctoral thesis focussed on Heisenberg inequality and uncertainty principles on locally compact groups, reflecting his deep engagement with analytical aspects of modern mathematics. Dr. Bansal's research interests lie primarily in harmonic analysis, uncertainty principles and the representation theory of locally compact and nilpotent Lie groups.
Over the course of his career, he has published a number of papers in well-regarded international journals and has presented his findings at multiple national and international conferences. His work in these areas has helped advance understanding of fundamental analytical structures and their applications in abstract mathematical contexts. A dedicated teacher with more than 16 years of experience in undergraduate education, Dr. Bansal has guided countless students through core mathematical concepts while emphasising clarity and rigour. In addition to classroom teaching, he has actively mentored students in research projects, encouraging them to explore mathematical ideas beyond standard curriculum.
