Problem Sheets in PDF (to appear): Sheet 1 Sheet 2 Sheet 3 Sheet 4

Course outline:

**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 l^{p} and L^{p} 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 l^{p} 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 L^{2} 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.

Wolfram Mathworld Pages on Functional Analysis

Wikipedia Page on Functional Analysis

Wikipedia Page on L

Mathematical discussions by Timothy Gowers

Old 321 web page for 2005-06 by Richard M. Timoney with Lecture Notes and Problem Sheets.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

I will appreciate any (also critical) suggestions that you may have for the course. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Feel free to come and see me if and when you have a question about anything in this course. Or use the feedback form from where you can also send me anonymous messages.