I'm trying to understand convenient vector spaces, but I'm unsure about the definition of the topology on smooth maps.


A map $f : E \rightarrow F$ between locally convex vector spaces $E$ and $F$ is called smooth iff it maps smooth curves to smooth curves. Set of all smooth maps is denoted with $C^\infty(E,F)$.


I'm confused which topology on $C^\infty(E,F)$ is used. In particular, to state Cartesian closedness of convenient vector spaces we need to make sense of $C^\infty(E, C^\infty(F, G))$. To use the definition of smooth maps we need a locally convex topology on $C^\infty(F, G)$.


I see three options for topology on $C^\infty(E,F)$:


1. Topology given by all seminorms in the form $p\_{q, x} = q(f(x))$, where $x\in E$ and $q\in \mathcal{P}\_F$(space of all seminorms on $F$)
2. The final topology of all curves $C^\infty(\mathbb{R}, C^\infty(E, F))$ where we understand $ C^\infty(E, F)$ as a locally convex space in the sense of (1.)   

This is kind of aligned with the definition of $c^\infty$ topology given [on wikipedia](https://en.wikipedia.org/wiki/Convenient_vector_space#The_%7F%27%22%60UNIQ--postMath-0000000D-QINU%60%22%27%7F-topology). But I'm confused by the sentence `In general, it is finer than the given locally convex topology, it is not a vector space topology, since addition is no longer jointly continuous.` If I understand it correctly, this topology does not give topological vector space, thus it does not produce locally convex space.
3. Topology given by all seminorms in the form $p\_{q,K,c} = \max\_{t \in K} (q\circ f \circ c)(t)$, where $q\in \mathcal{P}\_F$, $K\subset \mathbb{R}$ compact and $c \in C^\infty(\mathbb{R}, E)$. This is almost the definition from [The Convenient Setting of Global Analysis (3.11)](https://www.mat.univie.ac.at/%7Ekriegl/Skripten/apbook.pdf), there it is states as the initial topology given by precomposition with every smooth curve. Hopefully, I have managed to write it down more explicitly and correctly without using topology on $C^\infty(\mathbb{R}, F)$


Which one of these is used?



Convenient vector space and its locally convex structure
None of your suggestions is correct. Your third description is the closest, but you should include also finitely many of the derivatives there to reproduce KM 3.11 correctly, in a different form. A description of a fundamental system of seminorms for $C^\infty(E,F)$ can be given as follows: Take all seminorms $q\_{n,C}:C^\infty(E,F)\owns f\mapsto \sup\_{|t|,k\le n,c\in C}q\,({\rm D}^k(f\circ c)(t))$ where $n\in\mathbb Z^+$ and $C\subset C^\infty(\mathbb R,E)$ is nonempty and finite and $q$ is in some fundamental system of seminorms for $F$. Of course one could more generally have here a general $c^\infty$-open set $U$ in $E$ in place of the whole space $E$.


**Note 1.** (added later 28.10.2020) Having checked some of my standard references, and not having there found the phrase "fundamental system of seminorms", I add that by that for a locally convex space $F$ I mean any set $\mathcal S$ of continuous seminorms for $F$ such that for any continuous seminorm $\nu$ for $F$ there are some $\nu\_1\in\mathcal S$ and $A\in\mathbb R^+$ with $\nu\le A\,\nu\_1$ and such that for any $\nu\_1,\nu\_2\in\mathcal S$ there is $\nu\in\mathcal S$ with $\sup\,\{\nu\_1,\nu\_2\}\le\nu$.


**Note 2.** (added later 28.10.2020) Developing ${\rm D}^k(f\circ c)(t)$ as a finite sum of terms ${\rm d}^lf(c(t))\langle\,{\rm D}^{n\_1}c(t)\,,\ldots\,{\rm D}^{n\_l}c(t)\,\rangle$ where $1\le l\le k$ and $\sum\_i n\_i=k$ one can prove that the spaces $C^\infty(U,F)$ of the *version of the infinite-dimensional calculus* that in some circles is referred to by the phrase "Michal−Bastiani" are continuously embedded by inclusion in the corresponding KM spaces $C^\infty(U,F)\_{\_{^{\rm KM}}}$ whenever $E$ and $F$ are "convenient" locally convex spaces in the sense of Kriegl and Michor and $U$ is an open set in the locally convex topology of $E$.


Above, the spaces $C^\infty(U,F)$ are given the weakest (locally convex) topology that makes all the *variation*s $\delta^k:C^\infty(U,F)\to C\,(U\times E^k,F)$ defined by $f\mapsto\delta^kf$ given by $\delta^kf:\langle\,x,u\_1,\ldots\,u\_k\,\rangle\mapsto{\rm d}^kf(x)\langle\,u\_1,\ldots\,u\_k\,\rangle$ continuous when the spaces $C\,(U\times E^k,F)$ are given the topology of uniform convergence on compact sets.


