nLab gaseous vector space

 Idea

A notion of vector space in condensed mathematics.

Peter Scholze, in an answer to David Roberts‘ question at Mathoverflow, says

For many (but definitely not all) applications to geometry over the real numbers, the gaseous real vector spaces work just as well, and their theory is much easier to get off the ground than liquid real vector spaces. (Roughly speaking, complex- or real-analytic spaces are fine with gaseous vector spaces, smooth manifolds not so much. The reason is that tensor products of spaces of holomorphic or real-analytic functions behave correctly under the gaseous tensor product, but tensor products of spaces of $C^\infty$-functions are only correct under the liquid tensor product.)

 Related concepts

• liquid vector space

• solid vector space?

 References

• Peter Scholze, Geometrization of the real local Langlands correspondence, notes for ARGOS seminar SS 24. (pdf)

See also at condensed mathematics