nLab
Fréchet space

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 Hausdroff 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 $\mathrm{TVS}$.

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 $L^p(ℝ)$ for $p<1$ is not a Fréchet space, because it is not locally convex.

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

On the other hand, the locally convex direct sum of a countable number of copies of $ℝ$ 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

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

If the limit exists and is 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

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

$$DP(f)h=\underset{t\to 0}{\lim }\frac{1}{t}(P(f+th)-P(f))$$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 $P$ is continuous differentiable or $C^1$.

A simple, but nontrivial example is the operator

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

$$P(f)≔ff\prime$$P(f) \coloneqq f f'

with the derivative

$$DP(f)h=f\prime h+fh\prime$$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 ${\int }_a^{b}f(t)dt\in F$. 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 ${\int }_a^{b}f(t)dt\in F$ such that

(i) for every continuous functional $\varphi$ we have $\varphi ({\int }_a^{b}f(t)dt)={\int }_a^b\varphi (f(t))dt$,

(ii) for every continuous seminorm $\parallel \cdot \parallel$ we have $\parallel {\int }_a^{b}f(t)dt\parallel \le {\int }_a^b\parallel f(t)\parallel dt$

(iii) integration is linear and

(iv) additive, i.e. ${\int }_a^{b}f(t)dt+{\int }_b^{c}f(t)dt={\int }_a^{c}f(t)dt$

There is a version of the fundamental theorem of calculus:

Theorem

If P is $C^1$ and $f+th\in \mathrm{Domain}(P)$ for $0\le t\le 1$, then

$$P(f+h)-P(f)={\int }_0^{1}DP(f+th)hdt$$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 $C^1$ then so is their composition $Q\circ P$ and

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

The first derivative $DP$ is a function of two variables, the base point $f$ and the direction $h$. Since $DP$ 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

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

We can iterate this procedure to define derivatives of arbitrary order, and thus the notion of smooth functions between Fréchet spaces.

References

• Wikipedia

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