6
$\begingroup$

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 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. Maybe (if possible) $W$ could have some algebraic structure (ideal ?).

$\endgroup$
4
  • 1
    $\begingroup$ To be precise, when you write $C^\omega(\Omega)$, do really mean the set of (real or complex?) valued real analytic functions (which is the usual convention) or do you mean the set of (complex valued) holomorphic functions, for which the usual notation would be $H(\Omega)$? If you mean the former, then there is no topological complement since the set is dense. And, if you really mean the usual topology of $C^\infty(\Omega)$, then $B\subset\Omega$ should be required to be relatively compact. Bounded generally gives a strictly stronger topology. $\endgroup$
    – TaQ
    Jul 29, 2017 at 20:36
  • 1
    $\begingroup$ No, it is not. If e.g. $\Omega$ is the open unit disk, then $B=\Omega$ satisfies your requirement of being bounded open, and you get uniform convergence on $\Omega$ which is not the usual topology of $C^\infty(\Omega)$. I mean dense in your "large" space $C^\infty(\Omega)$. $\endgroup$
    – TaQ
    Jul 30, 2017 at 8:23
  • $\begingroup$ O.K. thanks (I was thinking of $\Omega=\mathbb{C}$). Corrected ! $\endgroup$ Jul 30, 2017 at 12:49
  • $\begingroup$ [If you mean the former, then there is no topological complement since the set is dense.]---> No, I meant the complex valued (i.e. holomorphic) functions, I made the ranges precise in the question. $\endgroup$ Jul 30, 2017 at 12:59

1 Answer 1

6
$\begingroup$

$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.

$\endgroup$
5
  • $\begingroup$ For the sake of completeness, which "recent article of Vogt"? $\endgroup$
    – David Roberts
    Jul 31, 2017 at 8:11
  • 1
    $\begingroup$ Splitting of exact sequences of Fréchet spaces in the absence of continuous norms from 2004. $\endgroup$ Jul 31, 2017 at 8:19
  • $\begingroup$ @JochenWengenroth [The space of smooth functions is isomorphic to $s^\mathbb N$]---> Sorry, but what is $s$ ? $\endgroup$ Jul 31, 2017 at 8:45
  • $\begingroup$ Interesting. Thank you for your answer (+1), I must digest it together with the reference you gave. $\endgroup$ Jul 31, 2017 at 8:46
  • $\begingroup$ @JochenWengenroth Well ... I feel sort of outsider. It seems, from the paper by P. Domanski and D. Vogt, that $s$ is the space of rapidly decreasing sequences. $\endgroup$ Aug 1, 2017 at 3:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.