Analysis

Objective

Context

This is an introduction to topology and analysis intended for ENSAE and HEC students who have not taken an MP preparatory class. The level of this course nevertheless exceeds that of the MP program. It is divided into four main parts: Topology; Continuity; Completeness and compactness; Pre-Hilbertian spaces.

Evaluation:

The final grade for the course will be the average of the continuous assessment grade (50%) and the written final exam (50%).

The continuous assessment grade (CC) is made up of three elements, each out of twenty points: (i) the mid-term grade, (ii) the grade for attendance at lectures, which are compulsory, and (iii) the grade for participation in lectures. This is calculated as follows: 50% of the mid-term grade + 25% of the attendance grade + 25% of the maximum between the attendance grade and the mid-term grade.

The attendance grade is calculated according to the grid available on the school's intranet.

Planning

1. Metric spaces -General remarks, topology of a metric space, limits and continuity
2. Normed vector spaces -General remarks, examples, continuous linear applications, continuity of operations.
3. Compacity -General remarks, use of sequences, finite-dimension normed vector spaces, convexity and compacity.
4. Banach spaces -Complete spaces, examples of Banach spaces, series in a Banach space, Banach algebra..
5. Sequences and series of functions -Simple limit of a sequence or series, uniform convergence, integer series.
6. Hilbert spaces -Definitions, orthogonal projections, duality in Hilbert spaces, separation of convex parts, orthonormed families, Fourier series.

References

[1] D. Guinin et B. Joppin. Analyse MP. Bréal, 2004.

[2] F. Liret et D. Martinet. Analyse 2e année. Dunod, 2004.

[3] H. Queffélec. Topologie. Dunod, 2006.

[4] L. Schwarz. Analyse I. Théorie des ensembles et topologie. Hermann, 1997.

[5] G. Skandalis. Topologie et analyse 3e année. Dunod, 2004.