The concept of Fréchet manifold is a special case of that of infinite-dimensional manifold: In analogy to how a finite-dimensional smooth manifold is a manifold modeled on a Cartesian space $\mathbb{R}^n$ in CartSp, a Fréchet manifold is a manifold modeled on a Fréchet space, such as notably $\mathbb{R}^\infty$ (exmpl.).
The category of Fréchet manifolds is a full subcategory of that of diffeological spaces (prop. 1 below) hence of smooth sets (see here).
It is possible to define, analogous to the finite dimensional case, the notion of smooth functions between Fréchet spaces, see at Fréchet space – Differentiable and smooth functions. Therefore, the usual definition of smooth manifold carries over word by word:
A Fréchet manifold is a Hausdorff topological space with an atlas of coordinate charts taking their value in Fréchet spaces, such that the coordinate transition functions are all smooth functions between Fréchet spaces.
It is possible to generalize some concepts of differential geometry from the finite case to the Fréchet case, one has to be careful, however:
The dual of a Fréchet space that is not a Banach space is never a Fréchet space, therefore one cannot e.g. define both the tangent and the cotangent bundle as Fréchet manifolds. More serious is however
The existence and uniqueness theorems for ordinary differential equations fail in infinite dimensions, so that theorems depending on that from finite dimensional differential geometry cannot be transscribed to the infinite situation in general. It is possible to do this on a case by case basis however.
There are several definitions of tangent vectors that are equivalent in the finite dimensional setting, but may be different in infinite dimensions. Tangent vectors can be defined to be derivations on germs of functions (algebraic definition), or as equivalence classes of smooth curves (kinematic definition). For the time being we settle with the kinematic definition:
kinematic tangent vector
The kinematic tangent vector space of a Fréchet manifold $M$ at a point $p$ consists of all pairs $(p, c'(0))$ where $c$ is a smooth curve
As usual, the set of pairs $(p, c'(0)), p \in M$ forms a Fréchet manifold, the tangent bundle $TM$.
The last sentence makes use of the notion of vector bundle, which can be defined exactly as in the finite dimensional setting:
vector bundle
A Fréchet manifold $V$ is a Fréchet vector bundle over $M$ with projection $\pi$, if for every point $p \in M$ there are charts of $M$ and $V$ such that $V$ is mapped locally to $U \subset F \times G$ for Fréchet spaces $F, G$, the projection $\pi$ corresponds to the projection of $U \times G$ to $U$, and the vector space structure on each fibre is that induces by the vector space structure on $G$.
Since, as mentioned before, the dual space of a Fréchet space that is not a Banach space is itself not a Fréchet space, we cannot define the cotangent space canonically as the dual space of the tangent space. Instead we define it directly:
differential form
A differential form (a one form) $\alpha$ is a smooth map
where $TM$ is the tangent bundle.
We discuss how Fréchet manifolds form a full subcategory of that of diffeological spaces.
Define a functor
from Fréchet manifolds to diffeological spaces (and hence to smooth spaces and smooth stacks) in the evident way by taking for $X$ a Fréchet manifold for any $U \in$ CartSp the set of $U$-plots of $\iota(X)$ to be the set of smooth functions $U \to X$.
The functor $\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces$ is a full and faithful functor.
This appears as (Losik, theorem 3.1.1).
Let $X, Y \in SMoothManifold$ with $X$ a compact manifold.
Then under this embedding, the diffeological mapping space structure $C^\infty(X,Y)_{diff}$ on the mapping space coincides with the Fréchet manifold structure $C^\infty(X,Y)_{Fr}$:
This appears as (Waldorf, lemma A.1.7).
For $X,Y$ two smooth manifolds, such that in addition $X$ is compact, then the mapping space, i.e. the set of smooth functions $C^\infty(X,Y)$ is naturally a Fréchet manifold. Under the full subcategory inclusion of Fréchet manifolds into diffeological spaces and smooth sets (prop. 1) this coincides with the canonical mapping space formed there.
For example smooth loop space (i.e. for $X = S^1$ the cirlce?) are Fréchet manifolds.
For details on this see at manifold structure of mapping spaces.
(see also Dodson-Galanis-Vassiliou 15)
Frecht manifolds may be thought of as projective limits of Banach manifolds (see the “added remark” at the end of this MO comment)
The infinite product Fréchet space $\mathbb{R}^\infty$ (exmpl.) is of course a Fréchet manifold.
As a Fréchet manifold, $\mathbb{R}^\infty$ (example 2) should be the projective limit
formed in the category of Fréchet manifolds.
The point that needs checking is that for $X$ any Fréchet manifold, then a continuous function
is smooth as soon as all its components
are smooth. This is checked for instance in (Saunders 89, lemma 7.1.8).
Conversely:
A function
out of $\mathbb{R}^\infty$ (example 2) is differentiable precisely if at each point only a finite number of its partial derivative are non-vanishing.
As a global generalization of the pro-finite dimensional Fréchet manifold $\mathbb{R}^\infty$ of example 2, every infinite jet bundle $J^\infty E = \underset{\longleftarrow}{\lim}_k J^k E$ is a Fréchet manifold, modeled on $\mathbb{R}^\infty$ (Saunders 89, chapter 7).
Beware, that infinite jet bundles are also naturally thought of as pro-manifolds. This differs from the Frechet manifold structure of example 3:
A morphism of pro-manifolds
is equivalently a function that is “globally of finite order”, in that there exists $k \in \mathbb{N}$ and an ordinary smooth function $f_k \colon J^k E \to \mathbb{R}$ such that $f = f_k \circ p_k$.
But by prop. 4 a morphisms of Fréchet manifolds
is only restricted to have finite order of partial derivatives at every point.
This is a weaker condition. In fact it seems to be also weaker than the condition of being “locally of finite order” considered in Takens 79. (The function $f$ is locally of finite order if for every point in $J^\infty E$ there exists a $k \in \mathbb{N}$ and an open neighbourhood $U_k$ of its image in $J^k E$ and a smooth function $f^U_k \colon U_k \to \mathbb{R}$ such that restricted to the pre-image of $U_k$ in $J^\infty E$ the function $f$ given by $f_k \circ p_k$).
Accounts include
Peter Michor, Manifolds of differentiable mappings, Shiva Publishing (1980) pdf
Andreas Kriegl, Peter Michor: The convenient setting of global analysis, AMS (1997) (pdf)
V. I. Arnold, B. A. Khesin, Topological methods in hydrodynamics. (Springer 1998, ZMATH)
Boris Khesin, Robert Wendt, The geometry of infinite-dimensional groups. (Springer 2009, ZMATH)
The embedding into diffeological spaces is due to
and reviewed in section 3 of
The preservation of mapping spaces under this embedding is due to
Fréchet manifold structure on jet bundles is discussed in
David Saunders, chapter 7 of The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.
C. T. J. Dodson, George Galanis, Efstathios Vassiliou,, p. 109 and section 6.3 of Geometry in a Fréchet Context: A Projective Limit Approach, Cambridge University Press (2015)