This book grew out of lectures given since 1964 at Yale University, the University
of Illinois at Chicago, and Bar Ilan University. The material covers the
usual topics of Measure Theory and Functional Analysis, with applications to
Probability Theory and to the theory of linear partial differential equations.
Some relatively advanced topics are included in each chapter (excluding the first
two): the Riesz–Markov representation theorem and differentiability in Euclidean
spaces (Chapter 3); Haar measure (Chapter 4); Marcinkiewicz’s interpolation
theorem (Chapter 5); the Gelfand–Naimark–Segal representation theorem
(Chapter 7); the Von Neumann double commutant theorem (Chapter 8); the
spectral representation theorem for normal operators (Chapter 9); the extension
theory for unbounded symmetric operators (Chapter 10); the Lyapounov Central
Limit theorem and the Kolmogoroff ‘Three Series theorem’ (Application I); the
Hormander–Malgrange theorem, fundamental solutions of linear partial differential
equations with variable coefficients, and Hormander’s theory of convolution
operators, with an application to integration of pure imaginary order (Application
II). Some important complementary material is included in the ‘Exercises’
sections, with detailed hints leading step-by-step to the wanted results. Solutions
to the end of chapter exercises may be found on the companion website for this
text: http://www.oup.co.uk/academic/companion/mathematics/kantorovitz.