There are different generalizations of the concept of differentiable manifold (usually of finite dimension) to the case with infinite dimension. Essentially, it is sought that these manifolds with infinite dimension are homeomorphic, as these topological spaces, to vector spaces of infinite dimension, and this gives rise to the following types of manifolds of infinite dimension:
I would like to know what their hierarchy of generality and suitability for applications. For example, I understand that every Hilbert manifold is a Banach manifold (though not the other way around), but I can't see the inclusions between the other types, or what specific advantages each type of manifold provides for applications. For example I conjecture this hierarchy:
$\text{Hilbert M.} \subset \text{Banach M.} \subset \text{Fréchet M.}$
Is that correct?
An (infinite dimensional) topological manifold is a (usually Hausdorff) topological space $(X,\tau)$ together with an atlas of charts $\{(\mathcal{U}_{\alpha},u_{\alpha},E_{\alpha})\}_{\alpha \in A}$ where $\{U_{\alpha}\}_{\alpha \in A}$ is an open cover of $X$, $E_{\alpha}$ a topological vector space (TVS) called the modeling space and a homoemorphism $u_{\alpha}:\mathcal{U}_{\alpha}\to E_{\alpha}$ such that whenever $\mathcal{U}_{\alpha\beta}\doteq \mathcal{U}_{\alpha}\cap\mathcal{U}_{\beta}\neq \emptyset$, the mapping $$ u_{\alpha\beta}\doteq u_{\alpha}\circ u_{\beta}^{-1}: u_{\beta}(\mathcal{U}_{\alpha\beta})\to u_{\alpha}(\mathcal{U}_{\alpha\beta}) $$ is a smooth diffeomorphism.
Generally you have to choose the modeling space: Hilbert, Banach, Fréchet, LCS, ecc...(Hilbert is the stronger notion, Locally Convex Spaces the weaker on the list); as well as a calculus (for general TVS there exists more that one choice). Of course the model space that you choose depends on the application you have in mind, but beware that the "nicer" your model space is, the more constrained the manifold structure becomes.
Example 1): By Theorem 1 in https://www.researchgate.net/publication/267073293_On_the_differential_topology_of_Hilbert_manifolds you see that if $X$ is a separable manifold modeled on Hilbert spaces then it can only be an open subset of the model space, effectively making useless the manifold notion.
Example 2) (loss of derivative): Suppose you have the (trivial) Banach manifolds $C^{0}_{2\pi}$ of continuous periodic functions in $\mathbb{R}$ with period $2\pi$ with the supremum norm. Then define the translation operator $$ L:\mathbb{R}\times C^{0}_{2\pi}\to C^{0}_{2\pi}: (t,f)\mapsto L_t(f) $$ with $L_t(f)(x)=f(x+t)$. You can show that it is continuous , however the mapping $$ \tilde{L}: \mathbb{R} \to \mathcal{L}(C^{0}_{2\pi},C^{0}_{2\pi}), $$ valued into the Banach space of linear and continuous operators, is not! The reason being that a small horizontal translation can result in a arbitrary high jump for the function. To avoid this problem we might consider $$ L:\mathbb{R}\times C^{1}_{2\pi}\to C^{0}_{2\pi}: (t,f)\mapsto L_t(f) $$ then the extra regularity implies that $\tilde{L}: \mathbb{R} \to \mathcal{L}(C^{1}_{2\pi},C^{0}_{2\pi})$ is continuous. The price we have paid is to go from a more regular space to less regular one (from $C^{1}_{2\pi}$ to $C^{0}_{2\pi}$). This is what is called the loss of derivative. To avoid this problem you could want to use infinitely differentiable functions, i.e. define $$ L:\mathbb{R}\times C^{\infty}_{2\pi}\to C^{\infty}_{2\pi}: (t,f)\mapsto L_t(f), $$ however those are no longer Banach manifolds (you can show that $C^{\infty}_{2\pi}$ is actually Fréchet).
This phenomena also appears when you study Banach Lie gruops, in which case the situation is even worse: there is a theorem [see Omori, H. (1978). On Banach-Lie groups acting on finite dimensional manifolds. Tohoku Mathematical Journal, Second Series, 30(2), 223-250.] that states: if G is a Banach Lie group acting faithfully and transitively on a finite dimensional manifold M, then G has to be finite dimensional! (keep in mind the example of the space of diffeomorphism of an open subset $\Omega$ of $\mathbb{R}^n$ acting on $\Omega$, the theorem states that $\mathrm{Diff}(\Omega)$ cannot have a Banach structure.)
Finally let me give an example where the Fréchet space structure is not enought [see Michor, P. W. (1980). Manifolds of differentiable mappings (Vol. 3). Birkhauser; or go to the author webpage where you can find a copy]. Suppose we want to describe general manifolds of mappings, i.e. spaces like $C^{\infty}(M,N)$ with $M$, $N$ finite dimensional differentiable manifolds, then we are forced to use LCS (unless N is a vector space and you can get away with Fréchet). One has charts $\{\mathcal{U}_{f},u_{f},\Gamma^{\infty}_c(M\leftarrow f^{*}TN)\}_{f\in C^{\infty}(M,N)}$ where $\mathcal{U}_f$ is the space of functions $g\in C^{\infty}(M,N)$ which differ from $f$ on a compact subset, $\Gamma^{\infty}_c(M\leftarrow f^{*}TN)$ is the LCS (indeed it is a limit-Fréchet space) of smooth compaclty supported functions, and $u_f$ is defined as follows: take any Riemannian exponential $\exp$ on $N$, then $$ u_f(g)=\exp_{f}^{-1}(g)\in \Gamma^{\infty}_c(M\leftarrow f^{*}TN) $$ where $\exp_{f}^{-1}(g)(x)=\exp_{f(x)}^{-1}(g(x))\in T_{f(x)}N$.