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.


  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.


[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.