Prolongement

Prolongement is a term used in French to denote the extension or continuation of a mathematical object from a domain or subset to a larger domain, or its analytic continuation to a broader region. The exact meaning varies by context, but it generally expresses the idea of extending definitions, properties, or structures beyond their original scope.

In real analysis and topology, prolongement often refers to extending functions from a subset to a larger space. A classical result is the Tietze extension theorem: any continuous function defined on a closed subset of a normal topological space can be extended to a continuous function on the whole space without increasing its range. For differentiable functions, the Whitney extension theorem provides criteria to extend a function defined on a closed subset of Euclidean space with preserved smoothness up to a chosen order. A frequently used practical prolongement is extension by zero: a function defined on a subset E is extended to the ambient space by assigning zero outside E, a construction common in Lp spaces.

In complex analysis, prolongement commonly means analytic continuation: a holomorphic function defined on one domain can often be extended along paths to a larger domain where it remains holomorphic. Such continuations are constrained by singularities and branch points, leading to concepts like domains of holomorphy and the monodromy phenomenon.

In other areas of mathematics, prolongement appears in differential geometry and algebraic geometry as the extension of geometric objects (sections, tensor fields, connections) along maps or across bases, and in the study of how local data extend to global objects.

Overall, prolongement captures the broad idea of carrying mathematical data from a restricted setting to a more expansive one, preserving as much structure as possible.