Fréchet spaces are particularly well-behaved topological vector spaces (TVSes). Every Cartesian space is a Fréchet space, but Fréchet spaces may have infinite dimension. There is analysis on Fréchet spaces, yet they are more general than Banach spaces; as such, they are popular as test spaces for possibly infinite-dimensional manifolds: Fréchet manifolds.
Beware the clash ofterminology: a ‘Fréchet topology’ on a ‘Fréchet topological space’ is something 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.)
A basic example of a Fréchet space is , as a topological space the projective limit over the finite dimensional Cartesian spaces (example 4 below) . This is not a Banach space anymore, since it does not carry a compatible norm anymore (e.g. Saunders 89, p. 253). But it evidently does carry the functions for all , where is the defining projection and where is the standard norm on . While not norms, these composites are seminorms on , they only fail the condition that only the 0-vector has vanishing norm. A Fréchet space is equivalently a vector space equipped with a countable familiy of seminorms, with compatibility conditions modeled on this example. See def. 2 below.
There are various equivalent definitions of Fréchet spaces:
such that the set of all open balls of the form
is a base of neighborhoods of .
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 the category of topological vector spaces.
Every Banach space is a Fréchet space.
The Lebesgue space for is not a Fréchet space, because it is not locally convex.
(Beware that the same symbol “” is also used for the limit of the same sequence but with with discrete topology, what leads to a linearly compact vector space as well as for the direct sum/inductive limit of , which is different.)
Then a compatible countable family of seminorms on , according to def. 2, is given by . Hence equipped with these, becomes a Fréchet space.
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.
This follows from the statement paragraph 29.1 (7) in (Koethe), which is: The strong dual of a locally convex metrizable TVS is metrizable iff is normable.
See also (Saunders 89, p. 255).
Every complete locally convex topological vector space is the cofiltered projective limit of Banach spaces in the category of locally convex spaces. (Note that Fréchet spaces are additionally required to be metrisable, so this is more general.)
To see this, choose a base of the neighborhood filter of , consisting of convex, balanced and absorbing sets and let be Minkowski functional associated to . The Hausdorffification of is easily seen to be a Banach space and because is directed by reverse inclusion so is . It is straightforward to check that in the category of locally convex spaces. For details, see (Schaefer-Wolff 99, Chapter II.§5, page 51ff.
Now given that a Fréchet space admits a decreasing sequence of convex balanced and absorbing neighborhoods, it follows immediately that:
Conversely, any limit of a countable sequence of Banach spaces is a Fréchet space.
See (Schaefer-Wolff 99), Chapter II.§4, page 48f (as well as Theorem I.6.1, page 28).
See also example 4 below.
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 continuous, 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. This allows to define the concept of smooth Fréchet manifolds.
PlanetMath, Fréchet space
Wikipedia, Fréchet space
Gottfried Koethe: Topological Vector Spaces I
H. Schaefer, Manfred Wolff Topological Vector Spaces, Springer (1999)
Discussion of analysis on Fréchet spaces includes
Richard S. Hamilton: The Inverse Function Theorem of Nash and Moser (Bulletin (New Series) of the American Mathematical Society Volume 7, Number 1, July 1982)
Discussion in the context of jet bundle theory includes