Course Details for A.Y. 2017/2018

Course Content

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elemtary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
• Introduction to the theory of unbounded operators. Linear differential operators. Applications.
• Introduction to infinite-dimensional differential calculus and variational methods

Textbooks

• John K. Hunter, Bruno Nachtergaele, Applied Analysis , World Scientific.    
• H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations , Springer.    

Internet Resources

Course page updates

• A.Y. 2012/2013
• A.Y. 2013/2014
• A.Y. 2014/2015
• A.Y. 2015/2016
• A.Y. 2016/2017
• A.Y. 2017/2018
• A.Y. 2018/2019
• A.Y. 2019/2020