Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1001	Special Topics in Functional Analysis	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	1	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Prof. BRECKNER Wolfgang, Ph.D., breckner@math.ubbcluj.ro	
Aims	The aim is to deepen the students&#8217; knowledge of functional analysis acquired during the undergraduate studies. There will be presented subjects concerning the fundamental algebraic-topological structures of functional analysis: topological linear spaces and locally convex spaces.	
Content	1. Elements of linear space theory: balanced sets, convex sets, maximal linear subspaces, hyperplanes
2. Linear topologies: neighbourhood bases, operators between topological linear spaces, continuity of linear functionals, separation of convex subsets of a topological linear space by a closed hyperplane
3. Locally convex topologies: characterizations of locally convex spaces, separation of convex subsets of a locally convex space by a closed hyperplane, extremal sets, extreme points
4. Linear spaces in duality: polars, the bipolar theorem, the duality pair generated by a separated locally convex space, weak topologies, the closed range theorem
	
Bibliography	1. Muntean I.: Analiză funcţională. Universitatea &#8222;Babeş-Bolyai&#8221;, Cluj-Napoca, 1993
2. Precupanu T.: Spaţii liniare topologice şi elemente de analiză convexă. Editura Academiei Române, Bucureşti, 1992 
3. Schaefer H. H., Wolff M. P.: Topological Vector Spaces. Second edition. Springer Verlag, New York, 1999
4. Werner D.: Funktionalanalysis. Vierte, überarbeitete Auflage. Springer Verlag, Berlin &#8211; Heidelberg &#8211; New York, 2002
5. Zălinescu C.: Programare matematică în spaţii normate infinit dimensionale. Editura Academiei Române, Bucureşti, 1998
	
Assessment	The students&#8217; knowledge will be assessed through two compulsory written tests: one examination in the seventh week of the semester, for the subjects studied during the first 6 weeks, and the following examination within the period of the exams session, for the subjects studied during the weeks 7 to 14. The final grade will be the arithmetic mean of the obtained grades, rounded if this mean is not an integer. The students willing to improve this grade and those absent from both written tests could take a single written examination for the entire course, during the re-examination session.