As an engineer who has not learned very much modern algebra, I recently learned about "class" in the algebra sense. Then I remembered our professor calling the set of Fourier transformable functions "the Schwarz class" instead of "set" or some other word.
Is there any connection between the algebra usage of the word "class" and properties of the functions in the set "the Schwarz class"?
This is very likely "class" in the sense of "set satisfying a property" rather than the formal set theory and logic definition of class: most analysts have probably never met a proper class (one that isn't a set) "in the wild". (This is not the only place you see this in mathematics: think of equivalence classes, for example.)