| Metrical spaces |
ter |
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| Teaching Staff in Charge |
| Prof. MURESAN Marian, Ph.D., mmarian@math.ubbcluj.ro Prof. KOLUMBÁN Iosif, Ph.D., kolumban@math.ubbcluj.ro Prof. DUCA Dorel, Ph.D., dduca@math.ubbcluj.ro Assoc.Prof. LUPSA Liana, Ph.D., llupsa@math.ubbcluj.ro |
| Aims |
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Getting to know the topology of the metric spaces with emphasis on the finit dimensional space Rn. |
| Content |
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1. The metric space of real numbers. Basic properties of real numbers. Absolute value function and the distance function on R. Bounded sets. The infimum and the supremum of a real number set. The topological structure of R. Sequences in R and their convergence. Fundamental sequences and the completeness of the real axis. Series of real numbers. Criteria for establishing the convergence of a series of real numbers.
2. Metric spaces. The Euclidean n-dimensional space Rn. Inner product and norms in Rn. The metric space Rn. Sequences in Rn and their convergence. The completeness of Rn. Bounded sets in Rn. The concept of metric space. Topological space. The topology of a metric space. Sequences in a metric space and their convergence. The notion of a complete metric space. Compact sets and sequentially compact sets in metric spaces. The compactness characterization in Rn. The notion of a linear normed space. Sequences in a linear normed space. Banach spaces. Series in a Banach space and their convergence, the general criterion of Cauchy. Convergent series in norm. 3. Functions between topological spaces. Functions berween metric spaces and between linear normed spaces. The limit of a function defined between metric spaces. The limit of the composition of functions. Continuity of functions. Continuity of the composition of functions. Continuity of the limit function to a uniformly convergent sequence of functions. Similar properies for series of functions. The Banach space of bounded functions. Uniformy continuous functions, Lipschitz functions and Holder functions. The notion of contraction. Banach fixed point theorem. |
| References |
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l. Balazs M., Kolumban I.: Matematikai analizis, Dacia Konyvkiado, Kolozsvar- Napoca, 1978
2. Breckner W.W.: Analiza matematica. Topologia spatiului Rn, Universitatea Cluj-Napoca, 1985 3. Cobzas St.: Analiza matematica (Calculul diferential), Presa Universitara clujeana, Cluj-Napoca, 1997 4. Duca D.I., Duca E.: Culegere de probleme de analiza matematica, l, 2, Editura GIL, Zalau, 1996,1997 4. Marusciac I.: Analiza matematica I: Universitatea Babes-Bolyai, Cluj-Napoca, 1980 5. Siretchi Gh.: Calcul diferential si integral,I, II, Editura stiintifica si enciclopedica, Bucuresti, 1985. 6. ***: Analiza matematica, I, Ed. a V-a, Editura didactica si pedagogica, Bucuresti, 1980 |
| Assessment |
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Exam. |