Fréchet spaces are particularly well-behaved topological vector spaces (TVSes). Every Cartesian space is a Fréchet space, but there are also infinite-dimensional examples. It is possible to do calculus on Fréchet spaces, yet they are more general than Banach spaces; as such, they are popular as test spaces for possibly infinite-dimensional manifolds; see Fréchet manifold.
Note that a ‘Fréchet topology’ on a ‘Fréchet topological space’ is different; this just means that a topological space satisfies the separation axiom. (Like all Hausdorff TVSes, Fréchet spaces satisfy this axiom, but they have a good deal of additional structure and properties.)
We accept as an automorphism of Fréchet spaces any linear homeomorphism; in particular, the particular translation-invariant metric or countable family of seminorms used to prove that a space is a Fréchet space is not required to be preserved. More generally, the morphisms of Fréchet spaces are the continuous linear maps, so that Fréchet spaces form a full subcategory of .
Every Banach space is a Fréchet space.
The Lebesgue space for is not a Fréchet space, because it is not locally convex.
The direct product of a countable number of copies of is a Fréchet space. Let be the projection onto the first copies and let be a choice of norm on . Then a countable family of seminorms on is given by .
On the other hand, the locally convex direct sum of a countable number of copies of is not a Fréchet space.
The dual of a Fréchet space is a Fréchet space iff is a Banach space.
Reference: This follows from the statement paragraph 29.1 (7) in Gottfried Koethe: Topological Vector Spaces I, which is: The strong dual of a locally convex metrizable TVS is metrizable iff is normable.
It is possible to generalize some aspects of calculus to Fréchet spaces, for example the definition of the derivative of a curve is simply the same as in finite dimensions:
For a continuous path in a Fréchet space we define
If the limit exists and is continous, we say that is continuously differentiably or .
And just as in the finite dimensional case, we can define the partial derivative, or rather: the directional or Gâteaux derivative:
Let F and G be Fréchet spaces, open and a nonlinear continuous map. The derivative of at the point in the direction is the map
If the limit exists and is jointly continuous in both variables we say that is continuous differentiable or .
A simple, but nontrivial example is the operator
with the derivative
It is possible to generalize the Riemann integral to Fréchet spaces, too: For a continuous path on an interval in a Fréchet space we look for an element . It turns out that such an element exists and is unique, if we impose some properties of the integral known from the finite dimensional case:
There exists a unique element such that
(i) for every continuous functional we have ,
(ii) for every continuous seminorm we have
(iii) integration is linear and
(iv) additive, i.e.
There is a version of the fundamental theorem of calculus:
If P is and for , then
The chain rule is valid:
If P and Q are then so is their composition and
The first derivative is a function of two variables, the base point and the direction . Since is already linear in , we define the second derivative with respect to only:
second derivative The second derivative of in the direction is defined to be
It is a theorem that the second derivative, if it exists and is jointly continuous, is bilinear in .
We can iterate this procedure to define derivatives of arbitrary order, and thus the notion of smooth functions between Fréchet spaces.
Calculus on Fréchet spaces is nicely explained in this paper: