Set theory. Operations with sets, their families. Functions and maps. Equivalence relations and equivalence classes. Cardinal numbers. Partial and linear order relations. Cantor-Bernstein-Schroeder Theorem. Well-ordered sets, ordinals. Transfinite induction. Comparability of ordinals. Axiom of choice. Zorn's Lemma. Existence of algebraic bases in vector spaces.
Metric and normed Spaces. Definitions, main examples. Cauchy-Schwarz, Höder and Minkowski inequalities. Their integral forms. The spaces lp and Lp as normed vector spaces. Open and closed sets. Interior, exterior, boundary, closure. Dense and nowhere dense sets. Separable and complete metric spaces, examples. Cantor's Intersection and Baire's Category Theorems.
Topological Spaces. Definitions, examples. Discrete and indiscrete topology. Neighborhoods, convergent sequences, continous maps. Bases and subbases. First and second countable spaces. Hausdorff spaces. Induced topology on a subset. Product topology. Frechet spaces and their topology. Topological vector spaces.
Linear Functionals and Operators. Continuity and boundedness of linear functionals and operators, their equivalence. Operator/functional norm. Dual spaces, their completeness. Minkowski functional. Hahn-Banach Theorems for real and complex normed spaces. Existence of closed complements for finite-dimensional subspaces. Canonical inclusion into double dual and reflexivity. Duals for lp spaces. Weak and weak * topology. Open Mapping and Closed Graph Theorems. Uniform Boundedness Principle.
Hilbert Spaces. Scalar (inner) products in real and complex vector spaces. Pre-Hilbert and Hilbert spaces. Cauchy-Schwarz inequality. Orthonormal systems and orthonormal bases. Gram-Schmidt orthogonalization. Bessel's inequality. Criteria for completenss of orthonormal systems. Isometric isomorphisms between abstract Hilbert spaces and appropriate L2 spaces. Orthogonal complements. Riesz representation theorem. Adjoint, self-adjoint, unitary and compact operators.
Distributions. Spaces of test functions. Definition of a distribution. Continuous functions as distributions. Dirac delta function as a distribution. Operations with distributions: addition, multiplication by smooth functions, differentiation. Distribution solutions of linear partial differential equations.
For exam-related problems look in TCD past examination papers and Mathematics department examination papers.
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