nLab Fréchet space

Topics in Functional Analysis

Differential geometry

differential geometry

synthetic differential geometry

Fréchet spaces

Idea

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 $T_1$ separation axiom. (Like all Hausdorff TVSes, Fréchet spaces satisfy this axiom, but they have a good deal of additional structure and properties.)

Definition

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

Equivalently, a Fréchet space is a complete Hausdorff TVS whose topology may be given (as a gauge space) by a countable family of seminorms.

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 $TVS$.

Examples

Every Banach space is a Fréchet space.

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.

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

The direct product of a countable number of copies of $\mathbb{R}$ is a Fréchet space. Let $\pi_n \colon \prod_k \mathbb{R} \to \mathbb{R}^n$ be the projection onto the first $n$ copies and let $\|\cdot\|_n$ be a choice of norm on $\mathbb{R}^n$. Then a countable family of seminorms on $\prod_k \mathbb{R}$ is given by $v \mapsto {\|\pi_n(v)\|_n}$.

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

Properties

Fréchet spaces are barrelled and bornological.

The dual of a Fréchet space $F$ is a Fréchet space iff $F$ 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 $F$ is metrizable iff $F$ is normable.

Calculus

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:

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 continous, 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.

References

Calculus on Fréchet spaces is nicely explained in this paper:

• 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)

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 23, 2014 04:11:33 by \?_{\pi}? (221.188.136.129)