| Real analysis (2) |
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| Teaching Staff in Charge |
| Prof. NEMETH Alexandru, Ph.D., nemab@math.ubbcluj.ro Assoc.Prof. ANISIU Valer, Ph.D., anisiu@math.ubbcluj.ro |
| Aims |
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Basic selected topics in general topology and measure theory, solid background for other courses like functional analysis or differential geometry.
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| Content |
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1. GENERAL TOPOLOGY. Ordered sets, nets and filters. Characteriations of
topological properties using nets and filters. Separation axioms (T3,T4), Uryson's lemma. Product spaces, Tychonoff's theorem, metrizability of a countable product of topological spaces. Baire spaces, generic properties. Metrizability of topological spaces. Classification of topological spaces using Venn diagrams. 2. MEASURE THEORY. Convergence types for sequences of measurable functions: almost uniform convergence, a.e. convergence, convergence in measure, Egorov's and Riesz's theorems. L^p spaces, completeness, the density of continuous functions, the Hilbert space L^2. Fourier series, Diriclet and Fejer kernels, localization principle, pointwise and uniform convergence. Real measures, positive, negative and total variation, Radon-Nikodym theorem, Jordan decomposition. Measure and integral on product spaces, Fubini's theorem. Radon derivatives and absolute continuous functions. |
| References |
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1. V. Anisiu: Topologie si teoria masurii. Universitatea "Babes-Bolyai", Cluj-Napoca, 1995.
2. N. Boboc, Gh. Bucur: Masura si capacitate. Ed. Stiintifica si enciclopedica, Bucuresti, 1985. 3. B. Gostiaux: Exercises de mathematiques speciales, Tome 2, Presse Universitaire de France, 1997. 4. J. Kelley: General topology. Van Nostrand, Princeton, 1950. 5. P. Kree: Integration et theorie de la mesure. Une approche geometrique. Ellipses, Paris, 1997 6. W. Rudin: Real and complex analysis, McGraw Hill, New York, 1988 (exista traducere in limba romana) |
| Assessment |
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Exam. |