Mat6, Spring 2013

Integration and Fourier Theory

The course is shared by Horia Cornean and Morten Grud Rasmussen.

ECTS: 5 ECTS-point.

Prerequisites: one should have previously followed Analysis 1 and 2 on MAT3 and MAT4, and Geometry on MAT5.

Content: Integration of positive measurable functions; the monotone and dominated convergence theorems. The Lebesgue measure on Rn and its invariance. Product-measure, Tonelli and Fubini theorems. Hölder and Minkowski inequalities; Lebesgue-spaces and their completeness. Approximation by smooth functions. Convolution and approximation of identity. Fourier series, Riesz-Fischer theorem; orthogonal expansions in Hilbert spaces. The Fourier transform and the inversion theorem; Plancherel's identity.

Exam: oral examination, with intern censorship.

Grade: Pass/fail

One should read the indicated material in advance, and try to solve the given exercises beforehand. The book is wonderful, but rather concise and difficult to read.


  1. Walter Rudin: Real and Complex Analysis. McGraw-Hill Education, Third edition. 1987

  2. H.D. Cornean: Noter om bl.a. Banach, Brouwer, Schauder, Kakutani fikspunktsætninger, Hairy Ball, Jordans kurvesætning, Nash ligevægtspunkt for endelige spil. (link)

  3. H.D. Cornean: Notes for the course Operatorer i Hilbertrum (link)


Written by Horia Cornean.