nLab
Fréchet space

Context

Functional analysis

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Fréchet spaces

Idea

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 T 1T_1 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 =lim n\mathbb{R}^\infty = \underset{\longleftarrow}{\lim} \mathbb{R}^n, as a topological space the projective limit over the finite dimensional Cartesian spaces n\mathbb{R}^n (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 p n n n\mathbb{R}^\infty \overset{p^n}{\longrightarrow} \mathbb{R}^n \overset{\Vert -\Vert_n}{\longrightarrow} \mathbb{R} for all nn \in \mathbb{N}, where p np^n is the defining projection and where n{\Vert -\Vert}_n is the standard norm on n\mathbb{R}^n. While not norms, these composites are seminorms on \mathbb{R}^\infty, 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.

Definition

There are various equivalent definitions of Fréchet spaces:

Definition

A Fréchet space is equivalently a complete Hausdorff locally convex vector space that is metrisable. The metric can be chosen to be translation-invariant.

Definition

A Fréchet space is a complete Hausdorff topological vector space VV whose topology may be given (as a gauge space) by a countable family of seminorms, hence for which there exists a family of seminorms

n:V,n {\Vert - \Vert}_n \;\colon\; V \longrightarrow \mathbb{R} \;\;\; \,,n \in \mathbb{N}

such that the set of all open balls of the form

B ϵ (n)(x){yV|xy n<ϵ}forxV,ϵ>0,n B_\epsilon^{(n)}(x) \coloneqq \left\{ y \in V \,\vert\, {\Vert x-y \Vert}_n \lt \epsilon \right\} \;\;\;\;\;\;\;\; for \; x \in V\,, \epsilon \gt 0 \,, n \in \mathbb{N}

is a base of neighborhoods of xx.

Definition

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 TVSTVS of topological vector spaces.

Examples

Example

Every Banach space is a Fréchet space.

Example

If XX is a compact smooth manifold, then the space of smooth maps on XX is a Fréchet space. This can be extended to some non-compact manifolds, in particular when XX is the real line.

Example

The Lebesgue space L p()L^p(\mathbb{R}) for p<1p \lt 1 is not a Fréchet space, because it is not locally convex.

Example

Consider the direct product (as topological vector spaces) of a countable number of copies of the real line \mathbb{R}

Equivalently the projective limit (as topological vector spaces)

lim n n=lim( 2 1 0) \mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim}_n \mathbb{R}^n = \underset{\longleftarrow}{\lim} \left( \cdots \to \mathbb{R}^2 \overset{}{\to} \mathbb{R}^1 \to \mathbb{R}^0 \right)

over all Cartesian spaces via their canonical projection maps.

(Beware that the same symbol “ \mathbb{R}^\infty” is also used for the limit of the same sequence but with n\mathbb{R}^n with discrete topology, what leads to a linearly compact vector space as well as for the direct sum/inductive limit of 2 3\mathbb{R}\to \mathbb{R}^2\hookrightarrow\mathbb{R}^3\hookrightarrow\ldots, which is different.)

Write

π n: n \pi_n \colon \mathbb{R}^\infty \longrightarrow \mathbb{R}^n

for the induced projection maps onto the first nn copies and let n\|\cdot\|_n be the canonical norm on n\mathbb{R}^n.

Then a compatible countable family of seminorms on \mathbb{R}^\infty, according to def. 2, is given by vπ n(v) nv \mapsto {\Vert\pi_n(v) \Vert_n}. Hence equipped with these, \mathbb{R}^\infty becomes a Fréchet space.

On the other hand, the locally convex direct sum of a countable number of copies of \mathbb{R} is not a Fréchet space.

Properties

General

Fréchet spaces are barrelled and bornological.

Duals

The dual of a Fréchet space FF is a Fréchet space iff FF 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 FF is metrizable iff FF is normable.

See also (Saunders 89, p. 255).

As projective limits

Every complete locally convex topological vector space XX 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 {U α} αA\{U_{\alpha}\}_{\alpha \in A} of the neighborhood filter of 00, consisting of convex, balanced and absorbing sets and let p αp_{\alpha} be Minkowski functional associated to U αU_{\alpha}. The Hausdorffification X αX_{\alpha} of (X,p α)(X, p_{\alpha}) is easily seen to be a Banach space and because AA is directed by reverse inclusion so is X αX_{\alpha}. It is straightforward to check that X=limX αX = \underset{\longleftarrow}{\lim} X_{\alpha} 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:

Every Fréchet space is a sequential projective limit of Banach spaces.

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).

(from this math.stackexchange comment)

See also example 4 below.

Differentiable and smooth functions

It is possible to generalize some aspects of analysis (differential calculus) to Fréchet spaces (e.g. Michor 80, chapter 8, Saunders 89, p. 256).

For example the definition of the derivative of a curve is simply the same as in finite dimensions:

Definition

For a continuous path in a Fréchet space f(t)f(t) we define

f(t)=lim h01h(f(t+h)f(t)) f'(t) = \lim_{h \to 0} \frac{1}{h} (f(t + h) - f(t))

If the limit exists and is continuous, we say that ff is continuously differentiably or C 1C^1.

And just as in the finite dimensional case, we can define the partial derivative, or rather: the directional or Gâteaux derivative:

Definition

directional derivative

Let F and G be Fréchet spaces, UFU \subseteq F open and P:UGP: U \to G a nonlinear continuous map. The derivative of PP at the point fUf \in U in the direction hFh \in F is the map

DP:U×FG D P: U \times F \to G
DP(f)h=lim t01t(P(f+th)P(f)) D P(f) h = \lim_{t \to 0} \frac{1}{t} ( P(f + t h) - P(f))

If the limit exists and is jointly continuous in both variables we say that PP is continuous differentiable or C 1C^1.

A simple, but nontrivial example is the operator

P:C [a,b]C [a,b] P: C^{\infty}[a, b] \to C^{\infty}[a, b]
P(f)ff P(f) \coloneqq f f'

with the derivative

DP(f)h=fh+fh D P(f) h = f'h + f h'

It is possible to generalize the Riemann integral to Fréchet spaces, too: For a continuous path f(t)f(t) on an interval [a,b][a, b] in a Fréchet space FF we look for an element a bf(t)dtF\int_a^b f(t) d t \in F. 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:

Theorem

There exists a unique element a bf(t)dtF\int_a^b f(t) d t \in F such that

(i) for every continuous functional ϕ\phi we have ϕ( a bf(t)dt)= a bϕ(f(t))dt\phi(\int_a^b f(t) d t) = \int_a^b \phi(f(t)) d t,

(ii) for every continuous seminorm {\| \cdot \|} we have a bf(t)dt a bf(t)dt{\| \int_a^b f(t) d t \|} \leq \int_a^b {\| f(t) \|} d t

(iii) integration is linear and

(iv) additive, i.e. a bf(t)dt+ b cf(t)dt= a cf(t)dt \int_a^b f(t) d t + \int_b^c f(t) d t = \int_a^c f(t) d t

There is a version of the fundamental theorem of calculus:

Theorem

If P is C 1C^1 and f+thDomain(P)f + t h \in Domain(P) for 0t10 \leq t \leq 1, then

P(f+h)P(f)= 0 1DP(f+th)hdt P(f + h) - P(f) = \int_0^1 D P(f + t h) \;h \; d t

The chain rule is valid:

Theorem

If P and Q are C 1C^1 then so is their composition QPQ \circ P and

D[QP](f)h=DQ(P(f))DP(f)h D [Q \circ P](f) h = D Q(P(f)) \; D P(f) \; h

The first derivative DPD P is a function of two variables, the base point ff and the direction hh. Since DPD P is already linear in hh, we define the second derivative with respect to ff only:

Definition

second derivative The second derivative of PP in the direction kk is defined to be

D 2P(f)(h,k)=lim t01t(DP(f+tk)hDP(f)h) D^2 P(f) (h, k) = \lim_{t \to 0} \frac{1}{t} (D P(f + t k) h - D P(f) h)

It is a theorem that the second derivative, if it exists and is jointly continuous, is bilinear in (h,k)(h ,k).

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.

References

Discussion of analysis on Fréchet spaces includes

  • Peter Michor, chapter 8 of Manifolds of differentiable mappings, Shiva Publishing (1980) pdf

  • 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

  • David Saunders, chapter 7 of The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

Refinement to noncommutative geometry by suitable smoothed C-star-algebras is discussed in

  • Nikolay Ivankov, Unbounded bivariant K-theory and an Approach to Noncommutative Fréchet spaces pdf

Revised on September 6, 2016 11:53:56 by Ingo Blechschmidt (137.250.162.16)