BOOKS  
Curriculum Vitae
Books

Books in English Edited books Books in Danish
Functional Analysis: Entering Hilbert Space (2^{nd} ed.) This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, selfadjoint operators on separable Hilbert spaces. It exhibits a construction of the space of p^{th} power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Hölder and Minkowski. The L^{p}spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated. In this second edition, I have expanded the material on normed vector spaces and their operators presented in Chapter 1 to include proofs of the Open Mapping Theorem, the Closed Graph Theorem and the Hahn–Banach Theorem. The material on operators between normed vector spaces is further expanded in a new Chapter 6, which presents the basic elements of the theory of Fredholm operators on general Banach spaces, not only on Hilbert spaces. This requires that we develop the theory of dual operators between Banach spaces to replace the use of adjoint operators between Hilbert spaces. With the addition of the new material on normed vector spaces and their operators, the book can serve as a general introduction to functional analysis viewed as a theory of infinite dimensional linear spaces and linear operators acting on them.
Functional Analysis: Entering Hilbert Space This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, selfadjoint operators on separable Hilbert spaces. It exhibits a construction of the space of p^{th} power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Hölder and Minkowski. The L^{p}spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated.
Matematikkens Uendelige Univers Denne bog henvender sig til alle der ønsker et indblik i matematikkens mange bidrag til kultur, videnskab og samfund. Bogen er derfor tilrettelagt som en række små essays, hvor der tilstræbes en mere beskrivende og fortællende fremstilling end normalt i bøger med et matematisk indhold. Emnerne i bogen er mangfoldige og strækker sig fra beskrivelse af geometriske former og fænomener i omverdenen til diskussion af nogle af uendelighedsbegrebets mange aspekter. Store dele af bogen kræver ikke specielle matematiske forkundskaber, men nok en vis optagethed af faget. Bogen giver i glimt smagsprøver på matematikkens abstrakte idèverden, herunder et indblik i nogle af matematikkens nyeste landvindinger.
Fundamental Concepts in Modern Analysis Many advanced mathematical disciplines, such as dynamical systems, calculus of variations, differential geometry and the theory of Lie groups, have a common foundation in general topology and calculus in normed vector spaces. In this book, mathematically inclined engineering students are offered an opportunity to go into some depth with fundamental notions from mathematical analysis that are not only important from a mathematical point of view but also occur frequently in the more theoretical parts of the engineering sciences. The book should also appeal to university students in mathematics and in the physical sciences. 