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Introduction to Hilbert Spaces with Applications

Lokenath Debnath and Piotr Mikusiński
Publication Date: 
Number of Pages: 
[Reviewed by
Mihaela Poplicher
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This book represents a comprehensive treatment of Hilbert Spaces and Linear Operators on Hilbert Spaces, two of the most important topics in functional analysis. It also includes some “prerequisites” (a chapter on Normed Vector Spaces and one on The Lebesgue Integral) and many applications: Integral and Differential Equations, Generalized Functions and Partial Differential Equations, Mathematical Foundations of Quantum Mechanics, Wavelets and Wavelet Transforms, Optimization Problems.

The authors intended to provide a graduate-level textbook. I think mathematics instructors will find material for at least two courses, and mathematicians at all stages in their career will find many useful reference materials.

The book is very readable; it contains complete proofs, many examples and exercises. This is the third edition of the text, which is a testament to its usefulness. There are a few new sections (like one on Sobolev Spaces), and most sections have been updated and expanded from previous editions.

In summary, this is a very useful and good book and it can find a place in the library of anybody interested in functional analysis, particularly Hilbert Spaces and their applications.

Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is

 CHAPTER 1 Normed Vector Spaces
CHAPTER 2 The Lebesgue Integral
CHAPTER 3 Hilbert Spaces and Orthonormal Systems
CHAPTER 4 Linear Operators on Hilbert Spaces
CHAPTER 5 Applications to Integral and Differential Equations
CHAPTER 6 Generalized Functions and Partial Differential Equations
CHAPTER 7 Mathematical Foundations of Quantum Mechanics
CHAPTER 8 Wavelets and Wavelet Transforms
CHAPTER 9 Optimization Problems and Other Miscellaneous Applications