Convenient vector space and its locally convex structure 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 $C^\infty(E,F)$:

*

*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 $F$)

*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. 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.

*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), 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?
 A: 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 variations $\delta^k:C^\infty(U,F)\to C\,(U\times E^k,F)$ 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.
