Here is an answer which interpolates between those by Piero and Nati.
In his Ph.D. thesis Grothendieck introduced the notion of nuclear topological vector space. Essentially, it is the correct definition of "finite-dimensional-like" infinite dimensional space. Namely, all theorems in multilinear algebra and probability which hold for finite dimensional spaces also hold verbatim for nuclear spaces. Although this is speculation on my part, I suspect that one thing Grothendieck learned from Laurent Schwartz is how powerful a mathematical idea a good definition can be, e.g., the definition of distribution. However, this theory would have been rather limited were it not for the second definition of nuclear spaces that Grothendieck provided. He then went on to repeat this trick again and again in algebraic geometry, and the rest is history.