Spaces to discover
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.
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Working environments for mathematicians
The abstract notion of space in mathematics is very different from what intuition suggests. Depending on the type of structure available, we can work with concepts from algebra, analysis, or geometry.
The world of functions
Many problems are tackled in function spaces, particularly those involving approximation.
Structuring randomness
Bringing order to the unpredictable is the aim of the concept of a probability space. By gathering the possible outcomes of a random experiment and their chances of occurring into a coherent structure, we can turn probability into an exact science.
Conquering spaces
Sequences and functions can have limits. But the need arose to define these ideas abstractly. The emergence of set theory and algebraic structures led, in the early 20th century, to the concepts of metric, topological and normed spaces. Here is their fascinating story.