The course is divided in two parts and it aims to provide a systematic exposition of the abstract measure theory , with additions on the fundamental theorem of integral calculus, and to present the definitions and first results on normed spaces, Banach and Hilbert spaces, also discussing projections and abstract Fourier series. The theory is accompanied by examples and exercises.

Lectures and exercises in the classroom.

The exam consists of a written test of not more than 2 hours (during which it is not allowed the use of notes, texts, minicomputers, ...) plus oral examination. The result of the written test is not binding to participate in the oral examination and the success of the examination, but of course it is an important element of judgment for the final evaluation.

The basics of Mathematical Analysis 1, 2 and Linear Algebra are supposed to be known.

Measure theory. Lebesgue measure, sigma-algebras, measures, measurable functions, Lebesgue integral, theorems of passage to the limit under the integral, almost-everywhere and quasi-uniform convergences, convergence in measure.

Product measures, Fubini and Tonelli theorems. Real measures, Hahn decomposition, absolutely continuous measures, Radon-Nikodym theorem, functions of bounded variation, absolutely continuous functions and the fundamental theorem of calculus.

Normed spaces and Banach spaces: foundations of the theory. Subspaces. Linear continuous operators. Dual space. Numerous examples. L^p spaces with their properties: The Young, Hölder, Minkowski inequalities. Completeness.

Hilbert spaces, Riesz and projections theorems. Fourier series: decomposition theorems, complete orthonormal systems, Riesz-Fisher theorem. Fourier series in L ^ 2_T and completeness of the system exp (ikT). Convolutions with trigonometric polynomials and Fejer kernels.

G. Gilardi: Analisi 3, McGraw-Hill,
H. Brezis: Functional Analysis, Springer.