2. Operator Algebras:
Topologies on bounded operators in Hilbert spaces, elementary theory of C*-algebras: positive linear functionals, commutative algebras, Gelfand representation theorems, GNS construction; elementary theory of von Neumann algebras: von Neumann's double commutant theorem, Kaplansky's density theorem; geometry of projections, types of factors.
Literature:
R. V. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras (Volume I), Academic Press.
G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, 1979.

3. Matrix Analysis:
Eigenvalues, singular values, and eigenvectors of matrices, positive definite matrices, matrix norms, the majorization relation, operator monotone and operator convex functions, differentiation of matrix functions, nonnegative matrices.
Literature:
R. Bhatia, Matrix Analysis, Springer.

4. Complex Analysis:
Complex line integrals, Cauchy formulas, Liouville's theorem, maximum principle and its variants (harmonic functions, Phragmén-Lindelöf theorems), Morera's theorem, residue theorem and its applications, conformal mappings, Weierstrass product, Mittag-Leffler theorem, analytic continuation.
Literature:
J. Bak, D. J. Newman, Complex Analysis, Springer.
J. Duncan, The elements of complex analysis, Wiley. 1968
H. A. Priestley, Introduction to Complex Analysis, Oxford Sci. Pub., 1990.

5. Fourier Analysis:
Convergence of Fourier series: Dirichlet kernel, Fejér kernel, convergence criteria, divergence phenomena. Fourier coefficients. Bochner's theorem. Properties of the Fourier transform. Fourier analysis on locally compact Abelian groups: duality, structure theorem, multiplier problem.
Literature:
M. Rudin, Fourier Analysis on Groups (Chapters I-V), Intersciences Publ. Edwards, Fourier Series I.

6. Functional Analysis:
Topological vector spaces, Banach spaces, dual spaces, the Fourier transform and the space of rapidly decreasing functions, basics of distribution theory, compact operators, elementary theory of Banach algebras, bounded operators in Hilbert spaces, the spectral theorem, the Cayley transform, self-adjoint operators, operator semigroups.
Literature:
W. Rudin, Functional Analysis, John Wiley.
J. B. Conway, A Course in Functional Analysis, Springer.

7. Linear Systems:
Linearization of problems. The transition matrix. Matrix exponential functions and inhomogeneous linear differential equations. Periodic equations. Asymptotic behavior. Linear time-varying and time-invariant systems. Controllability. Observability. The weighting pattern and minimal realizations. The time-invariant case: frequency response. Realization theory, McMillan degree. Feedback. Positive linear systems. Application of the least squares method. Stability. Digital filters and linear systems. Hardy spaces. Approximation and interpolation. Hankel norm approximation and minimization. System reduction.
Literature:
R. W. Brockett, Finite Dimensional Linear Systems, John Wiley, 1970.
C. K. Chui and G. Chen, Discrete Optimization, Springer, 1997.

8. Approximation Theory:
Existence and uniqueness of best approximation. Approximation by linear operators, Korovkin's theorem, Bernstein operator. Lagrange interpolation. Uniform convergence of best approximation, Weierstrass-type theorems. Best approximation in integral norms, orthogonal polynomials. Interpolation and approximation with spline functions. Extremal properties of Chebyshev polynomials.
Literature:
M. J. D. Powell, Approximation Theory and Methods, Cambridge Univ. Press, 1988.
R. A. Devore, G. G. Lorentz, Constructive Approximation, Springer, 1991.

9. Numerical Methods:
Error analysis; direct and iterative solutions of linear systems of equations; approximate calculation of eigenvalues and eigenvectors; numerical solutions of nonlinear equations and systems of equations; interpolation; approximation in the least squares sense; numerical integration; numerical solutions of initial and boundary value problems for ordinary differential equations.
Literature:
K. Atkinson, W. Han, Theoretical Numerical Analysis, Springer, 2007.
D. Kincaid, W. Cheney, Numerical Analysis, American Mathematical Society, 2009.
A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer, New York, 2000.