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.

Planning

I. TOPOLOGY

1. Topology in Metric Spaces
   1. Distances and Metric Spaces
   2. Open Sets in a Metric Space
   3. Topologies and General Topological Spaces
   4. Neighborhoods of a Point in a Metric Space
   5. Separated Spaces
   6. Closed Sets
   7. Interior and Closure
   8. Dense Subsets
   9. Normed Vector Spaces
2. Sequences
   1. Convergent Sequences
   2. Subsequences
   3. Characterization of Sets Using Sequences in Metric Spaces
   4. Distance to a Subspace
   5. Comparison of Topologies on the Same Space
3. Construction of Topological Spaces
   1. Subspace Topology
   2. Product Spaces

II. CONTINUITY

4. Definition and Characterizations

4.1 Definition and Examples

4.2 Characterizations

4.3 Continuous Linear Maps

5. Operations on Continuous Functions

5.1 Usual Operations

5.2 Sequences of Functions

III. COMPLETENESS AND COMPACTNESS

6. Cauchy Sequences and Complete Spaces

6.1 Cauchy Sequences and Completeness

6.2 Relationship Between Completeness and Closedness

7. Examples

7.1 Classical Examples

7.2 Fixed Point Theorem

7.3 Series

8. Compactness

8.1 The Concept of Compactness

8.2 Characterizations of Compactness in Metric Spaces

8.3 Continuous Functions on Compact Sets

8.4 Examples of Compact Spaces

8.5 Additional Results

IV. PRE-HILBERT SPACES

9. Definition

9.1 Inner Product on a Real Vector Space

9.2 Inner Product on a Complex Vector Space

10. Orthogonality

10.1 Definition and Examples of Applications

10.2 Gram-Schmidt Orthonormalization Process

10.3 Orthonormal Bases of Finite-Dimensional Vector Subspaces

11. Orthogonal Projection

11.1 Orthogonal Complement of a Subset

11.2 Orthogonal Complements and Orthogonal Projections

11.3 Orthogonal Projection onto a Finite-Dimensional Vector Subspace

11.4 Distance from a Vector to a Finite-Dimensional Subspace

11.5 Bessel's Inequality

11.6 Fourier Series

12. Hilbert Spaces

12.1 Definition and First Results

12.2 Projection onto a Closed Convex Set

12.3 Orthogonality in Hilbert Spaces

12.4 Hilbert Bases

13. Series of Functions

13.1 Modes of Convergence

13.2 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.