Bachelor's seminars (Measure Theory, Linear Operators in Hilbert Spaces, and related topics)
List of titles of 108 theses
(91 bachelor's and 17 master's theses) defended under my supervision at the Faculty of Applied Mathematics, AGH University of Krakow:
Bachelor's theses:
1. Existence and uniqueness of the Haar measure (2010),
2. Riesz representation theorem for linear functionals on the space of continuous functions with compact support (2010),
3. Applications of the Fourier transform (2010),
4. The Poisson process as an example of a point process (2010),
5. Convergence in measure spaces (2010),
6. Fubini's theorem and its applications (2010),
7. Absolute continuity of measures. The Radon-Nikodym theorem (2010),
8. Lebesgue measurable sets (2010),
9. Hausdorff measures (2010),
10. Square-integrable functions (2010),
11. Extensions of finitely additive functions to nonnegative measures (2010),
12. Spectral measure and its properties (2010),
13. The dual space of the \(L^p\) space (2010),
14. The Banach-Tarski paradox (2010),
15. Inclusions between \(L^p\) spaces (2010),
16. Introduction to distribution theory (2011),
17. Compact operators and their properties (2011),
18. Weak topology of Hilbert spaces (2011),
19. Weighted shifts (2011),
20. Baire's theorem and its applications (2011),
21. Orthonormal bases and their properties (2012),
22. Spectrum of a linear operator in a Hilbert space (2012),
23. Generalization of Andô's dilation theorem and Parrott's example (2014),
24. Semi-inner product spaces (2014),
25. Introduction to Orlicz function spaces (2014),
26. Properties of partial isometries (2014),
27. Properties of the adjoint operator in a Hilbert space (2014),
28. Spectral properties of a linear operator acting in a Hilbert space (2014),
29. Numerical range of an operator acting in a Hilbert space (2014),
30. Commutant lifting of a subnormal operator (2014),
31. Orthonormal bases of \(L^2\) spaces (2015),
32. Invertibility of linear combinations of two idempotents (2015),
33. The Gelfand transform (2015),
34. Legendre polynomials as an example of orthogonal polynomials (2015),
35. Compact operators in Hilbert spaces (2015),
36. Selected properties of operator topologies in the space of bounded operators on a Hilbert space (2015),
37. Spectrum and resolvent set of a linear operator in a Hilbert space (2015),
38. Spectral properties of compact operators (2015),
39. Properties of Banach algebras and \(C^*\)-algebras (2015),
40. The Cayley transform and its properties (2015),
41. Spectral radius of a matrix and a bounded linear operator in a Hilbert space (2015),
42. Introduction to the theory of Krein spaces (2017),
43. Matrix representation of operators in Hilbert spaces (2017),
44. Functional calculus for self-adjoint operators (2017),
45. The relationship between the norm and the spectral radius of a matrix (2017),
46. Completion procedures for unitary, normed, and metric spaces (2017),
47. Adjoint of an unbounded linear operator in a Hilbert space (2017),
48. Selected properties of Hilbert-Schmidt operators (2017),
49. Sobolev spaces (2017),
50. Continuity of linear operators in normed spaces (2018),
51. Introduction to weak topology (2019),
52. Uniformly convex spaces (2019),
53. Ring theory in abstract algebra (2019),
54. Completeness of metric spaces (2019),
55. Reflexivity of \(L^p\) spaces (2019),
56. Separability of \(L^p\) spaces (2019),
57. Orthogonal projections in Hilbert spaces (2019),
58. Gauss's divergence theorem and Stokes' theorem and their applications (2019),
59. Introduction to elliptic partial differential equations (2019),
60. On selected applications of the Hahn-Banach theorem (2020),
61. Description of parts of the spectrum of selected linear operators (2020),
62. Construction of the integral with respect to a spectral measure (2020),
63. Baire's theorem as a tool in analysis (2020),
64. Functions satisfying the Hölder condition (2020),
65. Variational formulation of selected elliptic problems (2020),
66. Norm of a bounded linear operator (2020),
67. Selected properties of the Volterra operator (2020),
68. Set of eigenvalues of a nonlinear operator in a two-phase problem (2024),
69. Function spaces defined on metric spaces (2024),
70. Complexification of bounded operators in Hilbert spaces (2024),
71. Selected properties of harmonic functions (2024),
72. Completions of normed and unitary spaces (2024),
73. Polar decomposition theorem for matrices (2024),
74. Various proofs of the uniform boundedness principle (2024),
75. Selected properties of \(p\)-summable sequence spaces (2024),
76. Zabreiko's lemma and its applications (2025),
77. Self-adjoint operators and symmetric operators (2025),
78. Wold decomposition theorem (2025),
79. Bases in Hilbert spaces (2025),
80. Applications of algebraic topology in proofs of Brouwer's fixed point theorem (2025),
81. Selected types of convergence in \(L^p\) spaces (2025),
82. Various representations of linear functionals (2025),
83. Selected properties of hyponormal operators (2025),
84. Selected criteria for subnormality of bounded operators (2026),
85. Probabilistic metric spaces (2026),
86. Selected properties of Polish spaces (2026),
87. Basic properties and applications of complex measures (2026),
88. Subnormality of a weighted shift operator (2026),
89. Closedness of the sum of closed subspaces of a Banach space (2026),
90. Construction of the Riemann-Stieltjes integral and its applications (2026),
91. Orlicz sequence spaces and their selected applications (2026);
Master's theses:
1. Ergodic properties of linear operators (2012),
2. Various proofs of the spectral theorem (2012),
3. General form of functionals on linear topological spaces (2013),
4. Various ways of defining topologies and their applications (2014),
5. Application of reproducing kernel Hilbert spaces in geodesy (2014),
6. Selected problems of dilation theory (2014),
7. Weighted shifts on graphs (2015),
8. Subnormality of generalized creation operators (2015),
9. Decompositions of pairs of commuting isometries acting in Hilbert spaces (2015),
10. Energy spaces of graph Laplacians (2016),
11. Selected problems of the spectral theory of non-self-adjoint operators (2017),
12. Extensions of operators acting in Hilbert spaces beyond the original space (2017),
13. Self-adjoint extensions of semibounded symmetric operators (2018),
14. Theorems on uniform ergodicity in Banach spaces (2020),
15. Existence of solutions for nonlinear equations in Hilbert spaces (2021),
16. Two-sided weighted shifts with operator-valued weights (2026),
17. Positive solutions of singular and non-autonomous \((p,q)\)-equations (2026).