Functional analysis - Part 1

The concepts presented in this presentation, the first part of a series dealing with functional analysis, relate more specifically to :

• finite-dimensional normed spaces: these are those for which an effective calculation, using the coordinates (in finite number!) of the vectors, is possible. From the point of view of functional analysis, they are characterized by the fact that they contain compact sets of non-empty interior: finite dimension and compactness are therefore intimately linked;

• Hilbert spaces; in particular, the separable Hilbert space is an analyst's paradise. It provides a natural framework for combining geometric ideas (orthogonality, Pythagorean theorem, etc.), algebraic ideas (eigenvalues, spectral theory, etc.) and analytic ideas (series and Fourier transforms);

• non-Euclidean Banach spaces; for example, the space of continuous functions or that of functions integrable over a segment are not Hilbert spaces. However, we need to consider them if we want to show the existence of solutions to differential equations, or develop the calculus of probabilities.

In the second part ([AF 101]), we will look at :

• non-normable functional spaces ;

• Fourier transformation ;

• probability calculus.

The knowledge required to approach this presentation of functional analysis requires familiarity with the basics of topology. These basics are presented in the article [AF 99] "Topology and Measurement" in this treatise.