Proportionality and geometry

From the beginning, proportionality, which is found in algebra with the rule of three, has become essential in geometry. It is encountered to prove the incommensurability of certain lengths or to evaluate areas and volumes. In Euclid, it is an indispensable tool for many proofs. Very closely linked to parallelism configurations, through the famous Thales theorem, it gave rise to famous results, such as Ceva's or Menelaus' theorems. Less known, antiparallelism and the associated ratios opened the way to new approaches. We owe to them the power of a point with respect to a circle or the inversion transformation.

Hilbert spaces, Banach spaces, normed vector spaces, topological spaces, measurable spaces... Conceptualized at the end of the 19th century and the beginning of the 20th, spaces are indispensable to the work of today's mathematician. We can imagine them as 'working environments' that provide 'tools' (distances, norms, scalar products, measure...), allowing us to formalize the notions involved in solving problems (convergence, limit, continuity...). We then discover that ideas initially appearing on sets of numbers or vectors can apply to more complex sets, like spaces of functions. A small guide to find one's way in spaces.