The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results about smooth maps from a euclidean space to a locally convex spaces.

The definition of smoothness for maps $U \to V$ where $U$ is $c^\infty$-open in a lctvs $W$, and $V$ is a lctvs, is given so that smooth maps are automatically the same as between maps between the associated diffeological spaces.

However, there is a notion of smoothness for maps between lctvs (due to Michal [1,2] and Bastiani [3]) not relying on K&M's definition, but I don't how they relate (Keller [4] treats this, I think, but I don't have access to that book).

Is it still true that lctvs embed into diffeological spaces using Michal-Bastiani smoothness?

[1] Michal, A. D. Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. USA 24 (1938), 340-342

[2] Michal, A. D. Differential of functions with arguments and values in topological abelian groups, Proc. Nat. Acad. Sci. USA 26 (1940), 356–359.

[3] Bastiani, A., Applications differentiables et varietes differentiables de dimension infinie, J. Anal. Math. 13 (1964), 1–114. (sorry for missing accents!)

[4] Keller, H. H., Differential Calculus in Locally Convex Spaces, Springer-Verlag, 1974

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results about smooth maps from a euclidean space to a locally convex spaces.

The definition of smoothness for maps $U \to V$ where $U$ is $c^\infty$-open in a lctvs $W$, and $V$ is a lctvs, is given so that smooth maps are automatically the same as between maps between the associated diffeological spaces.

However, there is a notion of smoothness for maps between lctvs (due to Michal [1,2] and Bastiani [3]) not relying on K&M's definition, but I don't how they relate (Keller [4] treats this, I think, but I don't have access to that book at the moment).

Is it still true that lctvs embed into diffeological spaces using Michal-Bastiani smoothness?

[1] Michal, A. D. Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. USA 24 (1938), 340-342 (pdf)

[2] Michal, A. D. Differential of functions with arguments and values in topological abelian groups, Proc. Nat. Acad. Sci. USA 26 (1940), 356–359. (pdf)

[3] Bastiani, A., Applications différentiables et variétés différentiables de dimension infinie, J. Anal. Math. 13 (1964), 1–114. (article, paywalled)

[4] Keller, H. H., Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics 417, Springer-Verlag, 1974 (Springerlink, paywalled)

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results about smooth maps from a euclidean space to a locally convex spaces.

The definition of smoothness for maps $U \to V$ where $U$ is $c^\infty$-open in a lctvs $W$ and $V$ is a lctvs is given so that smooth maps are automatically the same as between maps between the associated diffeological spaces.

However, there is a notion of smoothness for maps between lctvs not relying on K&M's definition (due to Michal [1,2] and Bastiani [3]), but I don't how they relate (Keller [4] treats this, I think, but I don't have access to that book).

Is it still true that lctvs embed into diffeological spaces using Michal-Bastiani smoothness?

[1] Michal, A. D. Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. USA 24 (1938), 340-342

[2] Michal, A. D. Differential of functions with arguments and values in topological abelian groups, Proc. Nat. Acad. Sci. USA 26 (1940), 356–359.

[3] Bastiani, A., Applications differentiables et varietes differentiables de dimension infinie, J. Anal. Math. 13 (1964), 1–114. (sorry for missing accents!)

[4] Keller, H. H., Differential Calculus in Locally Convex Spaces, Springer-Verlag, 1974

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results about smooth maps from a euclidean space to a locally convex spaces.

The definition of smoothness for maps $U \to V$ where $U$ is $c^\infty$-open in a lctvs $W$, and $V$ is a lctvs, is given so that smooth maps are automatically the same as between maps between the associated diffeological spaces.

However, there is a notion of smoothness for maps between lctvs (due to Michal [1,2] and Bastiani [3]) not relying on K&M's definition, but I don't how they relate (Keller [4] treats this, I think, but I don't have access to that book).

Is it still true that lctvs embed into diffeological spaces using Michal-Bastiani smoothness?

[1] Michal, A. D. Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. USA 24 (1938), 340-342

[2] Michal, A. D. Differential of functions with arguments and values in topological abelian groups, Proc. Nat. Acad. Sci. USA 26 (1940), 356–359.

[3] Bastiani, A., Applications differentiables et varietes differentiables de dimension infinie, J. Anal. Math. 13 (1964), 1–114. (sorry for missing accents!)

[4] Keller, H. H., Differential Calculus in Locally Convex Spaces, Springer-Verlag, 1974