# Some books by Chaim Samuel Hönig

Chaim Samuel Hönig published both textbooks aimed at undergraduates or postgraduates and research monographs. We list below a short selection of his books together with extracts from reviews.

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Aplicações da topologia à análise (1961)

Análise functional e aplicações. Vols. I, II. (1970)

Introdução às funções de uma variável complexa (1971)

Volterra Stieltjes-integral equations (1975)

1. Aplicações da topologia à análise (1961), by Chaim Samuel Hönig.
Review by: G S Young.
Mathematical Reviews MR0133796 (24 #A3621).

An approach to general topology emphasising especially the applications to analysis, by studying the Baire, Brouwer, Stone-Weierstrass, Ascoli theorems and the method of successive approximations, usually in a general setting, but often with less than maximum generality. These are applied to typical situations, such as integral equations, differential equations, compact hermitian operators, etc. Treatments are all quite elementary, with a certain elegance.
2. Análise functional e aplicações. Vols. I, II. (1970), by Chaim Samuel Hönig.
Review by: J B Prolla.
Mathematical Reviews MR0461068 (57 #1054).

The aim of this text is to familiarise the reader with the most important chapters of functional analysis, and to present significant applications of the theory. The exposition of the material covered is excellent. There are many examples and applications, and more than 300 exercises. The author restricts his attention to normed spaces. ... Volume I has two chapters; in one we find the elementary theory of Hilbert spaces and in the second the general theory of Banach spaces. In the first chapter the author presents the characterisation of inner-product norms, orthogonality and projections, orthonormal bases, the Riesz theorem on linear forms and the Lax-Milgram theorem on sesquilinear forms, and the Fourier transform on $L_{2}$. In the second chapter we find the duality theory and Hahn-Banach theorem, the Banach-Steinhaus theorem and application to Fourier series, the open mapping and the closed graph theorem and its applications. In Volume II we find the standard theory of linear operators. The text opens with preliminary material on compact metric spaces, Ascoli's theorem and compact operators. After that the elementary spectral theory of compact Hermitian operators in Hilbert space is developed and applied to integral and differential equations, in particular to the Sturm-Liouville problem. There is also the discussion of the spectral theory of bounded Hermitian operators. The treatment of the elementary theory of normed algebras is found in Section 9. In the final two sections the student is acquainted with the Riesz theory for compact operators and the Dirichlet problem for second order elliptic operators.
3. Introdução às funções de uma variável complexa (1971), by Chaim Samuel Hönig.
Review by: Editors.
Mathematical Reviews MR0499969 (58 #17705).

The book is a carefully written elementary introduction to complex variable theory. Chapter headings: Complex numbers; Series of functions in the complex plane; Complex differentiation; Complex integral; Calculus of residues.
4. Volterra Stieltjes-integral equations (1975), by Chaim Samuel Hönig.
Review by: James V Herod..
Mathematical Reviews MR0499969 (58 #17705).

The author presents results on the linear Volterra Stieltjes integral equation with linear constraints ... Some of the principal results contained in this monograph were announced previously ... The author begins the monograph by studying this integral and functions with values in a Banach space which are regulated - that is, the function have only discontinuities of the first kind. To illustrate the generality of the techniques, he shows that the integral equation may be realised as a linear differential equation, a Volterra integral equation, or a delay differential equation by appropriate choices of $K$. The constraint is identified as either initial conditions, boundary conditions, periodicity conditions, interface conditions or others. In this class of regulated functions, a resolvent for the integral equation and an integral representation for the linear constraint are determined. Necessary and sufficient conditions are given for there to be a Green function for the system.

Last Updated November 2022