Given complete, locally convex Hausdorff vector spaces $E$ and $F$, let
$$ E \otimes\_i F, \qquad E \otimes\_\pi F$$ denote the (completed) inductive and projective tensor products respectively. The inductive (resp. projective) tensor product has the universal property for separately (resp. jointly) continuous bilinear maps out of $E \times F$. By the universal property, there is a continuous linear map
$$E \otimes\_i F \to E \otimes\_\pi F.$$
If $E$ and $F$ are Frechet, then this map is an isomorphism. This is because separate and joint continuity coincide in this case. I am interested in knowing if this map is an isomorphism in situations where the hypotheses on $E$ are strengthened and the hypotheses on $F$ are relaxed.


More specifically, I am interested in the case $E = C^\infty(M)$, the nuclear Frechet space of smooth functions on a compact manifold $M$. In this case,
$$C^\infty(M) \otimes\_\pi F \cong C^\infty(M,F),$$ the space of smooth functions on $M$ with values in $F$. So I would like to know what hypotheses, if any, are needed on $F$ to ensure that
$$C^\infty(M) \otimes\_i F \to C^\infty(M,F)$$ is an isomorphism. For example, is it an isomorphism if $F$ is an $LF$-space, that is, an inductive limit of Frechet spaces?



Inductive tensor product and smooth functions
If you mean by LF-space a *strict* inductive limit of Frechet spaces (as it was done by Dieudonne and Schwartz) I think the answer is yes.


Here is what I believe could be made a proof: Since the inductive tensor product
respects inductive limits you have $C^\infty(M) \otimes\_i F = \lim C^\infty(M) \otimes\_i F\_n$ if $F= \lim F\_n$ is a strict LF-space. Since $F\_n$ is a closed topological
subspace of $F$ you get that $C^\infty(M) \otimes\_i F\_n = C^\infty(M) \otimes\_\pi F\_n$
and this implies that $C^\infty(M) \otimes\_i F = \lim C^\infty(M) \otimes\_\pi F\_n$ is a dense *topological* subspace of $C^\infty(M) \otimes\_\pi F$. On the other hand, it is complete
because it remains strict (as for as I remember this is due either to Dieudnne-Schwartz or to Köthe).




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EDIT: Sorry, I confused the Frechet case with the DF case -- only in the latter case
$\lim E \otimes\_\pi F\_n$ is a *topological* subspace of $E\otimes\_\pi \lim F\_n$.


For strict LF-spaces the answer to your question was already given in Grothendieck's thesis (part I, page 47): If E is a proper (non-normable) Frechet space and $F=\lim F\_n$ is a strict inductive limit with a *strictly* increasing sequence $F\_n$ then
$\lim E\otimes\_\pi F\_n$ is NEVER a *topological* subspace of $E \otimes\_\pi \lim F\_n$.


On the other hand, if the nuclear Frechet space $E$ has a continuous norm (as it is the case for $C^\infty(M)$) and $F=\lim F\_n$ is a strict LF-space then 
$\lim E\otimes\_\pi F\_n = E\otimes\_\pi F$ holds *algebraically*: 


The latter space is the space of all continuous linear operators $T$ from $E'$ to $F$ and the former of those operators with values in some $F\_n$. But $E'$ contains a bounded set
(the polar of the $0$-neighborhood corresponding to the continuous norm) which is total, i.e., its linear span is dense in $E'$. For any continuous $T$ the bounded set $T(B)$ is contained in some $F\_n$ (by the so called regularity of strict LF-spaces) and since $F\_n$ is closed in $F$ one gets from the continuity $T(E') \subseteq \overline{[T(B)]} \subseteq F\_n$.




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There might be non-strict inductive limits where $\lim E \otimes\_\pi F\_n = E\otimes\_\pi\lim F\_n$ holds topologically but I do not know an example since both, the algebraic and the topological, conditions are very restrictive. If $F$ is the dual of a nuclear Frechet space $X$ then the arguments above should show that the algebraic coincidence is equivalent to 
$L(X,E) = LB(X,E)$, the space of all operators from mapping some $0$-neighborhood of $X$ into a bounded subset of $E$. This situation has been investigated by Vogt,
J. Reine Angew. Math. 345 (1983), 182–200, and it implies Ext$^1(E,X)=0$. On the other
hand, I believe results of Grothendieck and Vogt yield that the topological coincidence
implies Ext$^1(X,E)=0$. If $X$ and $E$ are both power series spaces these conditions contradict each other.




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In conclusion: The topological equality $C^\infty(M) \otimes\_i F = C^\infty(M) \otimes\_\pi F$ seems very unlikely whenever $F$ is not a Frechet space. 


