# A Friendly Approach To Functional Analysis

Many books were written on this topic already. The current text can be called friendly because when the author gives a definition, he usually connects it to what the reader is supposed to know. These prerequisites are basically not much more than finite dimensional vector spaces. Since this is what functional analysis is about: calculus lifted to the level of infinite dimensional vector spaces. A lot of effort goes into pointing out the similarities and the differences. Since also Lebesgue integrals are used, an appendix introduces this concept in the one dimensional case.

But friendly does not mean that the reader is spared from effort. The text is peppered with many (197) exercises. The formulation of the exercises and the solutions provided at the end form an essential part of the book and they actually fill the larger part of its pages. Thus the reader who is new to the topic is supposed to work hard to properly grasp all the ideas. By including the solutions, the book is suitable for self study, although the text grew out of lecture notes used by the author while teaching at the London School of Economics.

On the other hand, the subject is vast, and since this is only an introduction not all the most difficult proofs are provided in full extent and not all the most complicated issues are discussed in all detail. It is important to mentioned that, even though the subject is abstract, attention is also paid to the application aspect. In fact the introduction starts with an optimal control problem as a motivation to embark on functional analysis. In a later chapter when differentiation and its application to optimality conditions is discussed, the Euler-Lagrange equation is derived and it is applied to classical Hamiltonian and Poissonian mechanics. This forms also the basis for a discussion of quantum mechanics in the chapter on Hilbert spaces. Compact operators are a reason to go into the subject of finite dimensional Galerkin approximations of the operators, which is important for numerical computations. Weak solutions of differential equations are obtained using distributions, discussed in the last chapter. This means that also physics and engineering students will appreciate this approach.

The text is rather dense and to the point and covers an enormous variety of topics, even though the main text has only six chapters packed on 258 pages. Here is a superficial sketch of the contents. The first chapter makes the step from vector spaces to normed spaces and to Banach spaces. The next obvious things to tackle in any calculus course are continuity and differentiation, which are also here the subjects of chapters 2 and 3. These include some operator theory with the open mapping theorem, some spectral theory and even a proof of the Hahn-Banach theorem. The application of the Fréchet derivative in optimization and the application in mechanics was mentioned above. The chapter on inner product and Hilbert spaces may be the easiest ones to deal with since these spaces behave mostly like finite dimensional vector spaces when separable, and they are frequently used in all kinds of applications: approximation, Gram-Schmidt orthogonalization, generalized Fourier series, problems involving self-adjoint operators, etc. The next chapter shows that to study operators, things become easier when we restrict ourselves to compact operators. The final chapter on distributions (to stress that it is only an introduction to the subject, its title is A glimpse of distribution theory) has as one of its applications the weak solutions of differential equations but also allows to extend Fourier analysis. A few of the topics that are considered to be more advanced (like the open mapping theorem, the dual space and the Hahn-Banach theorem, and the spectral theory of compact self-adjoint operators) are marked by a star, so that they can be skipped it needed.

This is a nice and modern introduction to the topic. The impatient or more advanced reader can just read the text, skipping the exercises (perhaps skip only their solutions, not all their statements since they often are formulations of additional properties to be used in further proofs), and so get a quick idea. Where possible, examples clarify some basics and some subtleties of the definition or the property. If, on the other hand, the reader is a student who wants to become proficient in the subject, then solving the exercises, or an many as possible, is an excellent way to acquire the necessary skills. The flexible possibilities provided —the text can be used as lecture notes for a course, or as a tool for self study, and even as a handbook to look up some definitions or theorems— is another of its great advantages.

**Submitted by Adhemar Bultheel |

**20 / Feb / 2018