Functional Analysis

mathematics

elective courses

This is a fundamental course in functional analysis. The course starts with basic notions on Banach and Hilbert spaces and their properties. The next topic of the course concerns linear functionals and operators in these spaces and their properties. The course gives also basic information on spectra and spectral oroperties of linear operators. Spectra of compact operators on Hilbert spaces are discussed.

1. The definition of a Banach space, sequence spaces, the L^p spaces, space C(K) and their completeness. Holder's and Minkowski's inequalities. The notion of a linear functional and its norm, examples.

2. Hilbert spaces, orthonormal bases and orthonormal sets of vectors, examples. The orthogonal projection and the characterization of linear continuous functionals on Hilbert spaces.

3. The notion of a linear operator and its norm. Examples of important linear operators: conditional mean. The Radon-Nikodym Theorem. The Fourier transform and Plancherel Theorem.

4. Adjoint operators on Banach and Hilbert spaces. Diagonalization of selfadjoint compact operator on a Hilbert space.

5. The Banach - Steinhaus Theorem and its applications, the Hahn-Banach Theorem and the separation theorems.

6. Possibly, the dual space, the space dual to L^p., C(K). The closed graph theorem and the open mapping theorem.

1. J.B. Conway, A course in functional analysis, Springer-Verlag 1985.

2. Y. Eidelman, V. Milman, A. Tsolomitis, Functional analysis. An introduction, AMS Graduate Studies in Mathematics, Vol. 66, American Mathematical Society, Providence 2004.

3. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler, Banach space theory, Springer 2011.

4. R.E. Megginson, An introduction to Banach space theory, Springer 1998.

5. W. Rudin, Real and complex analysis, McGraw-Hill Education, 1986

6. W. Rudin, Functional Analysis, McGraw-Hill, 1991

1. The students know the definition of a Banach space and know its properties, they know the sequence spaces, space C(K), Lebesgue spaces L^p, Hoelder and Minkowski inequalities, the notion of a linear functional and its norm.

2. The students know the definition of a Hilbert space and know its properties, they know the notion of an orthonormal set and an orthonormal basis. They know: the orthonormal projection theorem, examples of orthonormal bases: the trigonometric system, Haar system, wavelets, the form of linear functional on a Hilbert space.

3. The students know the definition of a linear operator and know its properties and the operator norm, examples of important operators, e.g., conditional mean operator and the Radon-Nikodym theorem, Fourier transform and Plancherel theorem.

4. The students know the definition of an adjoint operator on a Hilbert space and know their properties. They know the unitary operators. The diagonalization of a compact and self-adjoint o operator theorem.

5. The students know the Banach-Steinhaus Theorem and its applications, the Hahn-Banach separation Theorem.

6. The students know the definition of a dual space to a Banach space and their properties.

In particular the know the dual of C(K) and the Lebesgue L^p spaces, an adjoint operator on a Banach space. They have the preliminary knowledge about the weak and weak-star convergence. They know the closed graph and open mapping theorems.

7. The students are able to conduct mathematical reasoning: proving theorems as well as disproving conjectures and creating counterexamples.

Social competence:

1. The students understand the significance of the functional analysis as an abstract tool in other fields of Mathematics.

2. The students is able to use the language and methods of functional analysis in problems of Analysis and its applications.

The final grade based on number of points gained during classes, the midterm exam and the final exam.