ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Analysis

Objective

This course introduces concepts from topology in an analytical form: metric spaces, compacity, normed vector spaces and Banach spaces, together with several geometrical concepts: convexity and Hilbert spaces, including the statement and demonstration of projection theorems. A chapter also looks at sequences and series of functions. The course provides the essential foundations for differential calculus, optimisation and the theory of integration.

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. Convexity -Affine varieties, convex sets, convex functions.
  4. Compacity -General remarks, use of sequences, finite-dimension normed vector spaces, convexity and compacity.
  5. Banach spaces -Complete spaces, examples of Banach spaces, series in a Banach space, Banach algebra..
  6. Sequences and series of functions -Simple limit of a sequence or series, uniform convergence, integer series.
  7. 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. Dunod, 2004.