Number of lectures per week: 3

Assessment: several assignments, providing 10% of the credit for the course

End-of-year Examination: One 3-hour examination

Description: See http://www.maths.tcd.ie/~dwilkins/Courses/221/ for more detailed information.

Section 1: Sets Functions and Countability.

Sets; relations; functions between sets; injective, surjective and bijective functions; inverse functions; countability.

Section 2: Metric Spaces.

Euclidean spaces; definition of a metric space; definition and basic properties of convergence and limits for sequences of points in a metric space; definition and basic properties of continuity for functions between metric spaces; open and closed sets in metric spaces; continuous functions and open and closed sets; homeomorphisms.

Section 2: Analysis in Euclidean Spaces.

Euclidean spaces; definition and basic properties of convergence and limits for sequences of points in Euclidean spaces; definition and basic properties of continuity for functions between subsets of Euclidean spaces; uniform convergence; open and closed sets in Euclidean spaces.

Section 3: Complete Metric Spaces, Normed Vector Spaces and Banach Spaces.

 
the Least Upper Bound Principle; convergence of bounded monotonic sequences of real numbers; upper and lower limits; Cauchy's Criterion for Convergence; the Bolzano-Weierstrass Theorem; complete metric spaces; normed vector spaces; bounded linear transformations; spaces of bounded continuous functions; the Contraction Mapping Theorem; Picard's Theorem; the completion of a metric space.

Section 4: Topological Spaces.

Topological spaces; Hausdorff spaces; subspace topologies; continuous functions between topological spaces; homeomorphisms; sequences and convergence; neighbourhoods, closures and interiors; product topologies; cut and paste constructions; identification maps and quotient topologies; connectedness.

Section 5: Compact Spaces.

Definition and basic properties of compactness; compact metric spaces; the Lebesgue Lemma; uniform continuity; the equivalence of norms on a finite-dimensional vector space.

Section 6: The Extended Real Number System.

The extended real line; summation of countable sets; summable functions.

Section 7: Measure Spaces.

Lebesgue outer measure; outer measures; measure spaces; Lebesgue measure on Euclidean spaces; basic properties of measures.

Section 8: The Lebesgue Integral.

Measurable functions; integrals of measurable simple functions; the definition of the Lebesgue integral; Lebesgue's Dominated Convergence Theorem; the relationship between the Lebesgue and Riemann integrals.

Section 9: Signed Measures and the Radon-Nikodym Theorem.

Signed measures; the Hahn Decomposition Theorem; the Jordan Decomposition of a Signed Measure; Absolute continuity of measures; the Radon-Nikodym Theorem.

Oct 3, 2007