Let $E$ be locally convex topological vector space. Let $c^\infty E$ denote the same
vector space equipped with the $c^\infty$-topology (i.e. the finest topology on it, s.t.
all smooth curves $\mathbb{R} \rightarrow E$ are continous, see "The convenient setting
of global analysis" by A.Kriegl and P.Michor, I.2.12). If now $E,F$ are two such spaces,
it may happen that the identity $c^\infty(E \times F) \rightarrow c^\infty(E) \times
c^\infty(F)$ is continous but not a homeomorphism (see I.4.16 of the same reference).


I am wondering whether it is a homeomorphism in the following situation: Let $M$ be a smooth, finite-dimensional (but not necessarily compact) manifold and $V = V\_1 \oplus V\_2$ the Whitney sum of two finite-rank vector bundles over $M$. Then we have $\Gamma\_c(V) \cong \Gamma\_c(V\_1) \times \Gamma\_c(V\_2)$ as vector spaces and I think this is also an isomorphism of locally convex spaces if all these spaces are equipped with the usual strict inductive limit topology. But are the spaces $c^\infty \Gamma\_c(V\_1) \times c^\infty\Gamma\_c(V\_2)$ and $c^\infty\Gamma\_c(V)$ homeomorphic ? As far as I can see, they are linearly diffeomorphic (since the isomorphism maps smooth curves to smooth curves) but I don't know, under which (additional) conditions this map is also a homeomorphism. 



$c^\infty$-topologies on spaces of compactly supported sections and their products
$\def\Gmc#1{\Gamma\_{\rm c}(#1)}\def\ci{{\rm c}^\infty}
$Generally the linear isomorphism $\ci\Gmc{V\_1\oplus V\_2}\to\ci\Gmc{V\_1}\times\ci\Gmc{V\_2}$ is **not** a homeomorphism. For example, when taking $E=\mathcal D(\mathbb R)$, the identity map $\iota:c^\infty E\times c^\infty E\to c^\infty(E\times E)$ is not continuous. This follows from Proposition 4.26(ii) on page 45 in Kriegl and Michor's book. To see this, just observe that there it is shown that the vector addition $a:c^\infty E\times c^\infty E\to c^\infty E$ is not continuous. If $\iota$ were continuous, then $a$ would also be such since it is the composition $a\circ\iota:c^\infty E\times c^\infty E\to c^\infty(E\times E)\to c^\infty E$, and $a$ as a linear map *is* continuous $c^\infty(E\times E)\to c^\infty E$ as it is $E\times E\to E$.


**Added.** (in response to the comment, 23.8.2013) Applying the above argument to $E=s^{\kern.5mm(\mathbb N)}$ instead of $\mathcal D(\mathbb R)$ together with the existence of linear homeomorphisms $\Gmc{V\_i}\simeq s^{\kern.5mm(\mathbb N)}$ provided in the proof of Theorem 6.14 on page 73 in Kriegl and Michor's book, when $V\_1$ and $V\_2$ are non-zero-finite-rank (smooth) vector bundles over a non-compact but second countable non-zero-finite-dimensional manifold, from the diagram
$$\require{AMScd}
\begin{CD}
\ci\Gmc{V\_1\oplus V\_2} @>\simeq>> \ci(\Gmc{V\_1}\times\Gmc{V\_2}) 
 @>\simeq>> \ci(E\times E) \\
@. @| @| \\
{} @. \ci\Gmc{V\_1}\times\ci\Gmc{V\_2} @>>\simeq> \ci E\times\ci E \end{CD}
$$
one directly obtains the general result required.


**Remark 1.** There is a misprint in Kriegl and Michor's book at the end of the proof of Lemma 4.23 on page 44. Namely, there should be 



> 
> ... $b\_{N,K}\in U$ by (b''). Thus, ...
> 
> 
> 


in place of 



> 
> ... $b\_{N,K}\in U$ by (b'). Thus, ...
> 
> 
> 


The proofs are taken almost verbatim from Frölicher and Kriegl's book *Linear Spaces and Differentiation Theory*, and the same misprint occurs also there on page 194.


**Remark 2.** In Kriegl and Michor's Theorem 6.14, it is stated that the base manifold be *separable*. However, in view of the knowledge of existence of the two-dimensional smooth (or even analytic) [Gauld manifold](https://mathoverflow.net/questions/56275/is-a-connected-separable-locally-euclidean-hausdorff-topological-space-second-co/56333#56333) which is separable but not paracompact nor second countable, $\sigma$-compact, etc., I strongly suspect that this be sufficient. Rather, one should require the base to be e.g. second countable.


