Functional analysis. An elementary introduction (Q2921969)

Focusing on some material relevant in applied functional analysis such as variational methods on Hilbert spaces and model applications to differential and integral equations, the book provides basic notions in functional analysis without assuming any deep real analysis and without paying attention to several fundamental topics in functional analysis such as locally convex vector spaces, Banach algebras and weak topologies. The book of \textit{I. Gohberg} et al. [Basic classes of linear operators. Basel: Birkhäuser (2003; Zbl 1065.47001)] is close in spirit to this book. Each chapter includes suitable examples which make the book readable as well as a lot of exercises in three levels of difficulties. The book consists of the preface, 16 chapters including optional sections, some historical remarks, 6 appendices, a bibliography containing 50 references, and symbol, subject and author indices. The organization of the book is clear and rigorous but new for me: I see the author starts the book with inner product spaces and Fourier series without explicitly mentioning any topology for the notion of convergence and use the notation of norm while he will give the abstract definition of norm in the next chapter. In Chapter 2, he gives the notion of normed space and triangle inequality while he will introduce metric spaces in Chapter 3. Continuity is discussed in Chapter 4 and it is shown to be equivalent to the boundedness of linear mappings which have been studied in Chapter 3. In the sequel, Banach spaces, Hilbert spaces, Lebesgue spaces, Sobolev spaces and some operator theory are introduced. The Hahn-Banach theorem is presented in the last chapter. The book is suitable for students who need to study applied functional analysis and for non-specialists.