The formula you wrote for $G(x,y)$ doesn't seem to make sense, for it seems you're summing a scalar two-point distribution $\langle\Omega_1,\phi(x)\phi(y)\Omega_2\rangle$ with an operator-valued distribution $[\phi^{(+)}(x),\phi^{(-)}(y)]$ and equating the "result" to another, scalar two-point distribution $G(x,y)$.

@AymanMoussa your definition of (lc) inductive limit topology is correct, but that by itself doesn't guarantee that $E$ induces the topology of $E_n$ for all $n$. There are plenty of examples of (lc) inductive limits which are not strict.

@Math A closed subset $F$ of $E_n$ is closed in $E$ because it's a closed subset in the relative topology (for $E$ induces the topology of $E_n$ for all $n$ since it's a strict (lc) inductive limit), hence it's the intersection of $E_n$ (which is closed in $E$) with a closed subset of $E$.

With these provisos in mind, what Ayman shows is that the unit ball of $E_n$ is closed and bounded in $E$ but it cannot be compact there, for otherwise it would be compact in $E_n$ as well. Hence $E$ cannot be even semi-Montel, let alone Montel.

Not really, with the above remarks Ayman's proof is correct. Actually, the regularity of the inductive limit is not really needed (it only shows that any bounded subset of $E$ is a bounded subset of $E_n$ for some $n$, but we know that already for the unit ball of $E_n$ since continuous linear maps send bounded sets to bounded sets), but the closedness of $E_n$ in $E$ for all $n$ is directly needed to guarantee that a closed subset of $E_n$ is also closed in $E$.

Otherwise you cannot argue that if the unit ball of $E_n$ is compact in $E$, then it's compact in $E_n$ as well.

A thing missing here is the fact that $E=C^k_0(\Omega)$ is a strict (lc) inductive limit and therefore it induces on $E_n=C^k_{K_n}(\Omega)$ its original topology for all $n$. This is only true for strict (lc) inductive limits. Moreover, since in this case (also not generally true for all (lc) inductive limits) $E_n$ is closed in $E$ for all $n$, it follows that $E$ is a regular inductive limit, that is, any bounded subset of $E$ is contained and bounded in $E_n$ for some $n$. All that together implies that a compact subset of $E$ must be contained and compact in $E_n$ for some $n$.

@TobiasDiez true (and sorry for acknowledging this so late), I've added your remark to my answer.

One can apply the same procedure to uniquely improve the canonical Noether current associated to any local infinitesimal symmetry of first-order action functional(s), like local gauge symmetries (in the latter case, $X$ happens to be vertical). I don't know if this still works for higher-order Lagrangians, though. My Physics.SE answer physics.stackexchange.com/a/256496/16767 explains this in more detail.

There is a closely related problem which concerns the improvement of canonical Noether currents associated with symmetries of the action functional(s). In the case of the stress-energy tensor, for instance, $X$ is necessarily some lift of a vector field $X_M$ on $M$ and thus is never vertical. However, if one requires $X\rfloor\Theta$ to be ultralocal (that is, it depends only on point values of $X_M$ and not on the latter's derivatives), then at least for first-order Lagrangians this uniquely singles out $\Theta$.

The similarity of $R$ with the Nijenhuis tensor of an almost complex structure $J$ is due to the fact that the latter is half the Frölicher-Nijenhuis bracket of the vector-valued 1-form $J$ with itself, whereas $R$ is half the Frölicher-Nijenhuis bracket of the connection $\Phi$ with itself.