Abstract: Homological algebra of vector spaces is well understood. In functional analysis, many infinite dimensional vector spaces also contain analysis information. A diffeological vector space is a vector space with a compatible (generalized) smooth structure. In this talk, I will present a non-trivial example from functional analysis under the framework of diffeological vector spaces, see how the generalized smooth structure can be used to generalize a known result from analysis, as a motivation for the development of homological algebra of diffeological vector spaces. Then I will talk about the similarity and difference between this homological algebra and the homological algebra of R-modules. If time permits, some open questions will be discussed at the end.