Fréchet space

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.)

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 $TVS$.

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.

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.

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 $f(t)$ 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:

**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:

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:

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:

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:

**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**.

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
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