nLab Fréchet space

Idea

Fréchet spaces are particularly well-behaved topological vector spaces (TVSes). Every Cartesian space $\mathbb{R}^n$ is a Fréchet space, but Fréchet spaces may have non-finite dimension. There is analysis on Fréchet spaces, yet they are more general than Banach spaces; as such, they are popular as local model spaces for possibly infinite-dimensional manifolds: Fréchet manifolds.

A basic example of a Fréchet space is $\mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim} \mathbb{R}^n$, as a topological space the projective limit over the finite dimensional Cartesian spaces $\mathbb{R}^n$ (example 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 $\mathbb{R}^\infty \overset{p^n}{\longrightarrow} \mathbb{R}^n \overset{\Vert -\Vert_n}{\longrightarrow} \mathbb{R}$ for all $n \in \mathbb{N}$, where $p^n$ is the defining projection and where ${\Vert -\Vert}_n$ is the standard norm on $\mathbb{R}^n$. While not norms, these composites are seminorms on $\mathbb{R}^\infty$, they only fail the non-degeneracy condition saying that only the 0-vector has vanishing norm.

Generally, a Fréchet space is equivalently a real vector space equipped with a countable familiy of seminorms, with compatibility conditions modeled on this example. See def. below.

Beware the clash ofterminology: a ‘Fréchet topology’ on a ‘Fréchet topological space’ is something different; this just means that a topological space satisfies the $T_1$ separation axiom. (Like all Hausdorff topological vector spaces, Fréchet spaces satisfy this axiom, but they have a good deal of additional structure and properties.)

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

(via systems of seminorms)

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

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

such that the set of all open balls of the form

$$B_\epsilon^{(n)}(x) \coloneqq \left\{ y \in V \,\vert\, {\Vert x-y \Vert}_n \lt \epsilon \right\} \;\;\;\;\;\;\;\; for \; x \in V\,, \epsilon \gt 0 \,, n \in \mathbb{N}$$

is a base of neighborhoods of $x$.

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 $TVS$ of topological vector spaces.

Examples

Example

Every Banach space is a Fréchet space.

Example

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

Example

(the Schwartz space is a Fréchet space)

The Schwartz space $\mathcal{S}$ of functions with rapidly decreasing derivatives (this def.) is a Fréchet space.

See this prop.

Non-Example

The Lebesgue space $L^p(\mathbb{R})$ for $p \lt 1$ 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)

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

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 $\mathbb{R}^n$ with discrete topology, what leads to a linearly compact vector space as well as for the direct sum/inductive limit of $\mathbb{R}\to \mathbb{R}^2\hookrightarrow\mathbb{R}^3\hookrightarrow\ldots$, which is different.)

Write

$$\pi_n \colon \mathbb{R}^\infty \longrightarrow \mathbb{R}^n$$

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

Then a compatible countable family of seminorms on $\mathbb{R}^\infty$, according to def. , is given by $v \mapsto {\Vert\pi_n(v) \Vert_n}$. 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 topological vector space of a Fréchet space $F$ is itself again a Fréchet space precisely only if $F$ is in fact 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 $F$ is metrizable iff $F$ is normable.

See also (Saunders 89, p. 255).

As projective limits

Every complete locally convex topological vector space $X$ 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 $\{U_{\alpha}\}_{\alpha \in A}$ of the neighborhood filter of $0$, consisting of convex, balanced and absorbing sets and let $p_{\alpha}$ be Minkowski functional associated to $U_{\alpha}$. The Hausdorffification $X_{\alpha}$ of $(X, p_{\alpha})$ is easily seen to be a Banach space and because $A$ is directed by reverse inclusion so is $X_{\alpha}$. It is straightforward to check that $X = \underset{\longleftarrow}{\lim} X_{\alpha}$ 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 below.

Path smoothness

Definition

(path smooth linear function)

Let $V$ be a Fréchet vector space (def. ). Then a linear function

$$\mu \;\colon\; V \longrightarrow \mathbb{R}$$

is called path smooth if for every smooth function

$$g \;\colon\; \mathbb{R} \longrightarrow V$$

the composite

$$\mu \circ g \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}$$

is a smooth function.

Proposition

(path-smooth linear functions on a Fréchet space are continuous)

Let $V$ be a Fréchet vector space (def. ). Then every linear function on $V$ which is path-smooth (def. ) is continuous.

(Moerdijk-Reyes 91, chapter II, lemma 3.7)

Prop. implies for instance that distributions are the smooth linear functionals. See there for more.

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 $f(t)$ we define

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

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

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, $U \subseteq F$ open and $P: U \to G$ a nonlinear continuous map. The derivative of $P$ at the point $f \in U$ in the direction $h \in F$ is the map

$$D P: U \times F \to G$$

$$D P(f) h = \lim_{t \to 0} \frac{1}{t} ( P(f + t h) - P(f))$$

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

A simple, but nontrivial example is the operator

$$P: C^{\infty}[a, b] \to C^{\infty}[a, b]$$

$$P(f) \coloneqq f f'$$

with the derivative

$$D P(f) h = f'h + f h'$$

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

(i) for every continuous functional $\phi$ we have $\phi(\int_a^b f(t) d t) = \int_a^b \phi(f(t)) d t$,

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

(iii) integration is linear and

(iv) additive, i.e. $\int_a^b f(t) d t + \int_b^c f(t) d t = \int_a^c f(t) d t$

There is a version of the fundamental theorem of calculus:

Theorem

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

$$P(f + h) - P(f) = \int_0^1 D P(f + t h) \;h \; d t$$

The chain rule is valid:

Theorem

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

$$D [Q \circ P](f) h = D Q(P(f)) \; D P(f) \; h$$

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

Definition

second derivative The second derivative of $P$ in the direction $k$ is defined to be

$$D^2 P(f) (h, k) = \lim_{t \to 0} \frac{1}{t} (D P(f + t k) h - D P(f) h)$$

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

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

• Dietmar Vogt, Lectures on Fréchet spaces, 2000 (pdf)

• PlanetMath, Fréchet space

• Wikipedia, Fréchet space

• Gottfried Koethe: Topological Vector Spaces I

• H. Schaefer, Manfred Wolff Topological Vector Spaces, Springer (1999)

• Andreas Kriegl, Fréchet Spaces lecture notes, 2016 (pdf)

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 bundles and locally pro-manifolds includes

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

• Igor Khavkine, Urs Schreiber, section 2.2 of Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)

Discussion in the context of synthetic differential geometry includes

• Ieke Moerdijk, Gonzalo Reyes, around lemma 3.7 in chapter II of Models for Smooth Infinitesimal Analysis, Springer 1991

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