According to the authors’ statement in the preface, this book is intended to provide an introduction to the ideas and methods of linear functional analysis at a level appropriate to the final year of an undergraduate course. Only a basic knowledge of linear algebra and real analysis is assumed.
The following enumeration of the headings of the seven chapters may give an idea of the topics covered in the book: 1. Preliminaries (linear algebra, metric spaces, Lebesgue integration); 2. Normed Spaces (examples, finite-dimensional spaces, Banach spaces); 3. Hilbert Spaces (inner products, orthogonality, orthogonal complements, orthonormal bases, Fourier series); 4. Linear Operators (continuous linear transformations, the norm of a bounded linear operator, the space \(B(X,Y)\), dual spaces, inverses of operators); 5. Linear Operators on Hilbert Spaces (the adjoint of an operator, normal, self-adjoint and unitary operators, the spectrum of an operator, positive operators and projections); 6. Compact Operators (compact operators, spectral theory of compact operators, self-adjoint compact operators); 7. Integral and Differential Equations (Fredholm integral equations, Volterra integral equations, differential equations, eigenvalue problems and Green’s function).
The presentation is quite elementary, and there are sufficiently many illuminating examples and exercises (including solutions in the appendix). So this nice textbook perfectly fits the readership, i.e. undergraduate students in mathematics and physics, which the Springer series addresses to. It may be recommended to all students who want to get in touch with the basic ideas of functional analysis and operator theory for the first time.