**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.

**First semester:**

**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 Euclidean spaces.
**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; homeomorphisms.
**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 T

On 3 Oct 2007, 17:27.