Mathematical analysis 2

Metric spaces, functions between such spaces.

Complete and incomplete, separable and nonseparable metric spaces, Banach contraction principle.

Open, closed, bounded, compact and connected sets in metric spaces; properties of the families of open and closed sets, respectively. A characterization of continuous functions between metric spaces in terms of open sets.

Number series (definitions of such series, its convergence and divergence, criteria for convergence and divergence).

Function series (power and trigonometric series), radius and interval of convergence of power series.

Differentiation and integration of the limits of function sequences as well as of functions defined by function series.

Fourier series.

Normed and Banach spaces, linear bounded operators between Banach spaces, norms of such operators.

Inner and Hilbert spaces. Norms in Hilbert spaces, Schwarz inequality, parallelogram identity.

Fundamental knowledge and abilities on metric spaces and the importance of this knowledge for the unification of the analysis of functions of several variables, fundamental knowledge and abilities on series of real numbers, tests for their convergence and divergence, sequences and series of functions; their convergence and uniform convergence and tests on checking these facts, applications of function (power and trigonometric) series, Banach spaces and operators between such spaces.

Handbooks:

W. Rudin, Podstawy analizy matematycznej, PWN, Warszawa, 1982

H. J. Musielakowie, Analiza matematyczna, Vol.1, part 1 & 2, Wydawnictwo Naukowe UAM, Poznań, 1993

A. Sołtysiak, Analiza matematyczna, Part I & II, Wydawnictwo Naukowe UAM, Poznań, 1995(1996)

G. M. Fichtenholz, Rachunek różniczkowy i całkowy, Vol. 1,2 i 3, PWN, Warszawa 1994 (1995)

Bernd S.W. Schroder, Mathematical analysis, Wiley-Interscience, A John Wiley h Sons, Inc, Publication, 2008

Robert A. Adams, Calculus: a complete course, Addison-Wesley Publishers Limited, 1991

Exercises books:

W. Krysicki, L. Włodarski, Analiza Matematyczna w zadaniach, Vol. I, PWN, Warszawa 2003

J. Banaś, S. Wędrychowicz, Zbiór zadań z analizy matematycznej, Wydawnictwo Naukowo-Techniczne, Warszawa 2003

B.P. Demidowicz, Zbiór zadań z analizy matematycznej, Vol. I,II, Naukowa Książka, Lublin 1992(1993)

Wiesława J. Kaczor, Maria T. Nowak, Zadania z analizy matematycznej, Vol. I & II, PWN, Warszawa 2005(2005)