In the hope of completing the rich tapestry of complemented (or not) topological vector subspaces, I would like to know (maybe it is immediate for specialists) 
whether the space of analytic functions is complemented within the space of infinitely differentiable ones. I begin with the one-variable case 


... and make this precise. 



> 
> Let $\Omega\subset \mathbb{C}$ be an open subset. We consider  
> 
>  $$
> H(\Omega)=C^\omega(\Omega;\mathbb{C})\subset C^\infty(\Omega;\mathbb{C})
> $$
>  the large one being endowed with the standard topology defined by the seminorms
>  $$
> p\_{\,n,B}=sup\_{\ 0\leq |\alpha|\leq n\atop t\in B}|D^\alpha(f)[t]|\ . 
> $$
>  where $n\in \mathbb{N}, \alpha\in \mathbb{N}^2$, $B$ is a relatively compact open subset of $\Omega$ and the bi-indexed derivative is 
>  $$
> D^\alpha:=(\frac{\partial}{\partial x})^{\alpha[1]}(\frac{\partial}{\partial y})^{\alpha[2]}\ .
> $$ 
>  I know that the subspace $H(\Omega)=C^\omega(\Omega;\mathbb{C})$ is complete and then closed for this (standard) topology. My question is the following
> 
> 
> Q) Is there a known closed complement of it i.e. a decomposition
>  $$
> C^\infty(\Omega;\mathbb{C})=C^\omega(\Omega;\mathbb{C})\oplus W=H(\Omega)\oplus W
> $$  
> 
>  where $W$ is closed ? (maybe the projector is an integro-differential operator ?) at least for some particular domains $\Omega$ ?
> 
> 
> 


**Remark** i) This question is a reformulation of 
[this one](https://math.stackexchange.com/questions/2361195/complement-of-the-space-of-analytic-functions)
in MSE where it did not receive a complete answer. 


ii) With the given topology, $C^\infty(\Omega;\mathbb{C})$ and $H(\Omega)=C^\omega(\Omega;\mathbb{C})$ are m-convex [Fréchet algebras](https://en.wikipedia.org/wiki/Fr%C3%A9chet_algebra). Maybe (if possible) $W$ could have some algebraic structure (ideal ?).



Is this closed subspace of Fréchet space complemented
$H(\Omega) $ is not complemented in $C^\infty (\Omega)$ e.g. for the unit disc in $\mathbb C $. This follows from the structure theory of Frechet spaces: The space of smooth functions is isomorphic to $s^\mathbb N$ and has a certain property (DN$\_{loc}$) of Vogt. If $H (\Omega) $ were complemented it would also have this property and hence even property (DN) because it has continuous norms. But this is not true for power series spaces of finite type. 


I don't have any literature at hand. The book of Meise and Vogt is a good starting point. The local condition is in a more recent article of Vogt.


