The [nLab casually remarks](http://ncatlab.org/nlab/show/diffeological+space#References) 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](http://ncatlab.org/nlab/show/Boman%27s+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?
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[1] Michal, A. D. *Differential calculus in linear topological spaces*, Proc. Nat. Acad. Sci. USA **24** (1938), 340-342 ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077109/pdf/pnas01796-0040.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](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078188/pdf/pnas01616-0038.pdf))


[3] Bastiani, A., *Applications différentiables et variétés différentiables de dimension infinie*, J. Anal. Math. **13** (1964), 1–114. ([article](http://dx.doi.org/10.1007/BF02786619), paywalled)


[4] Keller, H. H., *Differential Calculus in Locally Convex Spaces*, Lecture Notes in Mathematics 417, Springer-Verlag, 1974 ([Springerlink](http://dx.doi.org/10.1007/BFb0070564), paywalled)



Do locally convex topological vector spaces embed into diffeological spaces?
The answer is *no*. In what follows (see OP's comments below) we assume that the arrows of the category of lctvs are Michal-Bastiani smooth maps. Recall that a map $\Phi:E\rightarrow F$ from a lctvs $E$ into another lctvs $F$ is said to be *Michal-Bastiani smooth* if its directional (Gâteaux) derivatives of order $k$


$$ D^k\Phi[x](y\_1,\ldots,y\_k) = \left.\frac{\partial^k}{\partial\lambda\_1\cdots\partial\lambda\_k}\right|\_{\lambda\_1=\cdots=\lambda\_k=0}\Phi\left(x+\sum^k\_{j=1}\lambda\_j y\_j\right)$$


exist and the maps $D^k\Phi:E\times E^k\rightarrow F$ are *(jointly) continuous* for all $k\in\mathbb{N}$. This notion of smoothness is the one used in Milnor's treatment of infinite-dimensional Lie groups (J. Milnor, "Remarks on Infinite-Dimensional Lie Groups". In: B. DeWitt, R. Stora, eds., Les Houches Session XL, *Relativity, Groups and Topology II* (North-Holland, 1984), pp. 1007-1057) and Hamilton's exposé of the Nash-Moser inverse function theorem (R.S. Hamilton, "The Inverse Function Theorem of Nash and Moser". [Bull. Amer. Math. Soc. **7** (1982) 65-222](http://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2/S0273-0979-1982-15004-2.pdf)).


Michal-Bastiani smooth maps are also smooth in the sense of Kriegl and Michor (i.e. composing with a smooth curve leads to another smooth curve; some call the latter *convenient smoothness*), since the chain rule holds for Michal-Bastiani smooth maps. The converse is *not* necessarily true - both notions of smoothness coincide for Fréchet spaces, but in general they *differ*: conveniently smooth maps need not even be continuous, whereas Michal-Bastiani smooth maps are always so. A counter-example to continuity for conveniently smooth maps may be found in the paper of H. Glöckner, "Discontinuous Non-Linear Mappings on Locally Convex Direct Products". Publ. Math. Debrecen **68** (2006) 1-13, [arXiv:math/0503387](http://arxiv.org/abs/math/0503387). In particular, if neither $E$ nor $F$ are Fréchet spaces, there may be maps between $E$ and $F$ which are smooth in the diffeological sense and need not even be continuous, let alone Michal-Bastiani smooth. Therefore, the natural inclusion functor of lctvs into diffeological spaces is not fully faithful. It is not clear from the above counterexample whether there are convenient *diffeomorphisms* which fail to be continuous or Michal-Bastiani smooth. (see however TaQ's answer below for such a counterexample)


