Can we define smooth diffeomorphisms on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature?

Question

Can we define smooth diffeomorphisms on the Hilbert cube$[0,1]^\mathbb{N}$ ? Has it been done in the literature?

For example, let $E$ be the set of bijections $f$ of the Hilbert cube, such that $f((x_i)_{i \in \mathbb{N}})= (f_n(x_1,...,x_n),x_{n+1},..)$ with $f_n$ a smooth diffeomorphism of $[0,1]^n$ for some integer $n$. The smooth distance on finite-dimensional smooth diffeomorphisms extends to $E$: $d(f,g)=d(f_{n},g_n)$ (we take $n$ to be the maximum of "dimensions" of $f$ and $g$). We can take the completion of $E$ for $d$. Has this space been considered before? If yes, where? If no, why?

Answer 1

$[0,1]^{\mathbb N}$ is not a manifold since it has too many corners. But it is a Frölicher space in the sense of section 23 of here. It is also a diffeological space in the sense of

• MR3025051
  Iglesias-Zemmour, Patrick(F-PROV-APT) Diffeology. Mathematical Surveys and Monographs, 185. American Mathematical Society, Providence, RI, 2013. xxiv+439 pp.

In both settings you can speak of diffeomorphisms, and yours fall among them.

Answer 2

An addendum to what wrote Peter Michor above: Frölicher spaces identify naturally with the full subcategory of reflexive diffeological spaces. (exercises 79 and 80 of the book). So, for your space equipped with the Frölicher structure, the two approaches coincide. That said I am curious to know why you are interested in this question?