School of Mathematics
School of Mathematics
Course 221 - Analysis 2007-08 (SF Mathematics,
SF Two-subject Moderatorship with Economics,
optional JS Two-subject Moderatorship
Lecturer: Dr. D.R. Wilkins and Dr. F. Jaeck
Duration: 24 weeks
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
See http://www.maths.tcd.ie/~dwilkins/Courses/221/ for more
- Section 1: Basic Theorems of Real Analysis.
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;
the Intermediate Value Theorem.
- 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
- Section 3: Metric 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;
- Section 4: Complete Metric Spaces, Normed Vector Spaces
and Banach Spaces.
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 5: 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 6: 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.
Second semester: measure theory; the Lebesgue integral.
Oct 3, 2007
File translated from
On 3 Oct 2007, 17:27.