Fréchet space explained
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces.
A Fréchet space
is defined to be a
locally convex metrizable topological vector space (TVS) that is
complete as a TVS, meaning that every
Cauchy sequence in
converges to some point in
(see footnote for more details).
[1] Important note: Not all authors require that a Fréchet space be locally convex (discussed below).
The topology of every Fréchet space is induced by some translation-invariant complete metric. Conversely, if the topology of a locally convex space
is induced by a translation-invariant complete metric then
is a Fréchet space.
Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a complete metrizable topological vector space, without the local convexity requirement (such a space is today often called an "F-space"). The local convexity requirement was added later by Nicolas Bourbaki. It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex. Moreover, some authors even use "F-space" and "Fréchet space" interchangeably. When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "-space" and "Fréchet space" requires local convexity.
Definitions
Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of seminorms.
Invariant metric definition
A topological vector space
is a
Fréchet space if and only if it satisfies the following three properties:
- It is locally convex.[2]
- Its topology be induced by a translation-invariant metric, that is, a metric
such that
for all
This means that a subset
of
is open if and only if for every
there exists an
such that
is a subset of
- Some (or equivalently, every) translation-invariant metric on
inducing the topology of
is complete.
- Assuming that the other two conditions are satisfied, this condition is equivalent to
being a complete topological vector space, meaning that
is a complete uniform space when it is endowed with its canonical uniformity (this canonical uniformity is independent of any metric on
and is defined entirely in terms of vector subtraction and
's neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on
is identical to this canonical uniformity).
Note there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.
Countable family of seminorms definition
The alternative and somewhat more practical definition is the following: a topological vector space
is a
Fréchet space if and only if it satisfies the following three properties:
- It is a Hausdorff space,
- Its topology may be induced by a countable family of seminorms
This means that a subset
is open if and only if for every
there exists
and
such that
\{v\inX:\|v-u\|k<rforallk\leK\}
is a subset of
- it is complete with respect to the family of seminorms.
A family
of seminorms on
yields a Hausdorff topology if and only if
A sequence
in
converges to
in the Fréchet space defined by a family of seminorms if and only if it converges to
with respect to each of the given seminorms.
Comparison to Banach spaces
In contrast to Banach spaces, the complete translation-invariant metric need not arise from a norm. The topology of a Fréchet space does, however, arise from both a total paranorm and an -norm (the stands for Fréchet).
Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the potential lack of a norm, many important results in functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold.
Constructing Fréchet spaces
Recall that a seminorm
is a function from a vector space
to the real numbers satisfying three properties. For all
and all scalars
If
, then
is in fact a norm. However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows:
To construct a Fréchet space, one typically starts with a vector space
and defines a countable family of seminorms
on
with the following two properties:
and
for all
then
;
is a sequence in
which is
Cauchy with respect to each seminorm
then there exists
such that
converges to
with respect to each seminorm
Then the topology induced by these seminorms (as explained above) turns
into a Fréchet space; the first property ensures that it is Hausdorff, and the second property ensures that it is complete. A translation-invariant complete metric inducing the same topology on
can then be defined by
The function
maps
monotonically to
and so the above definition ensures that
is "small" if and only if there exists
"large" such that
is "small" for
Examples
From pure functional analysis
- Every Banach space is a Fréchet space, as the norm induces a translation-invariant metric and the space is complete with respect to this metric.
- The space
of all real valued sequences (also denoted
) becomes a Fréchet space if we define the
-th seminorm of a sequence to be the
absolute value of the
-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.
From smooth manifolds
of all infinitely differentiable functions
becomes a Fréchet space with the seminorms
for every non-negative integer
Here,
denotes the
-th derivative of
and
In this Fréchet space, a sequence
of functions
converges towards the element
if and only if for every non-negative integer
the sequence
converges uniformly.
The vector space
of all infinitely differentiable functions
becomes a Fréchet space with the seminormsfor all integers
Then, a sequence of functions
converges if and only if for every
the sequences
converge compactly.The vector space
of all
-times continuously differentiable functions
becomes a Fréchet space with the seminormsfor all integers
and
If
is a compact
-manifold and
is a Banach space, then the set
of all infinitely-often differentiable functions
can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If
is a (not necessarily compact)
-manifold which admits a countable sequence
of compact subsets, so that every compact subset of
is contained in at least one
then the spaces
and
are also Fréchet space in a natural manner.As a special case, every smooth finite-dimensional
can be made into such a nested union of compact subsets: equip it with a Riemannian metric
which induces a metric
choose
and letLet
be a compact
-manifold and
a vector bundle over
Let
denote the space of smooth sections of
over
Choose Riemannian metrics and connections, which are guaranteed to exist, on the bundles
and
If
is a section, denote its jth covariant derivative by
Then(where
is the norm induced by the Riemannian metric) is a family of seminorms making
into a Fréchet space.From holomorphicity
- Let
be the space of entire (everywhere holomorphic) functions on the complex plane. Then the family of seminormsmakes
into a Fréchet space.
- Let
be the space of entire (everywhere holomorphic) functions of exponential type
Then the family of seminormsmakes
into a Fréchet space.
Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is the space
with
Although this space fails to be locally convex, it is an
F-space.
Properties and further notions
If a Fréchet space admits a continuous norm then all of the seminorms used to define it can be replaced with norms by adding this continuous norm to each of them. A Banach space,
with
compact, and
all admit norms, while
and
do not.
A closed subspace of a Fréchet space is a Fréchet space. A quotient of a Fréchet space by a closed subspace is a Fréchet space. The direct sum of a finite number of Fréchet spaces is a Fréchet space.
A product of countably many Fréchet spaces is always once again a Fréchet space. However, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if all for at most countably many of them are trivial (that is, have dimension 0). Consequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis). So for example, if
is any set and
is any non-trivial Fréchet space (such as
for instance), then the product
is a Fréchet space if and only if
is a countable set.
Several important tools of functional analysis which are based on the Baire category theorem remain true in Fréchet spaces; examples are the closed graph theorem and the open mapping theorem. The open mapping theorem implies that if
are topologies on
that make both
and
into
complete metrizable TVSs (such as Fréchet spaces) and if one topology is
finer or coarser than the other then they must be equal (that is, if
\tau\subseteq\tau2or\tau2\subseteq\tauthen\tau=\tau2
).
Every bounded linear operator from a Fréchet space into another topological vector space (TVS) is continuous.
There exists a Fréchet space
having a
bounded subset
and also a dense vector subspace
such that
is contained in the closure (in
) of any bounded subset of
All Fréchet spaces are stereotype spaces. In the theory of stereotype spaces Fréchet spaces are dual objects to Brauner spaces. All metrizable Montel spaces are separable. A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual converges is strongly convergent.
of a Fréchet space (and more generally, of any metrizable locally convex space)
is a
DF-space.
[3] The strong dual of a DF-space is a Fréchet space. The strong dual of a
reflexive Fréchet space is a
bornological space and a
Ptak space. Every Fréchet space is a Ptak space. The strong bidual (that is, the
strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.
Norms and normability
If
is a locally convex space then the topology of
can be a defined by a family of continuous on
(a
norm is a positive-definite
seminorm) if and only if there exists continuous on
Even if a Fréchet space has a topology that is defined by a (countable) family of (all norms are also seminorms), then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any single norm). The space of all sequences
(with the product topology) is a Fréchet space. There does not exist any Hausdorff
locally convex topology on
that is
strictly coarser than this product topology. The space
is not normable, which means that its topology can not be defined by any
norm. Also, there does not exist
continuous norm on
In fact, as the following theorem shows, whenever
is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to the presence of
as a subspace.
If
is a non-normable Fréchet space on which there exists a continuous norm, then
contains a closed vector subspace that has no topological complement.
A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.[4] In particular, if a locally convex metrizable space
(such as a Fréchet space) is normable (which can only happen if
is infinite dimensional) then its
strong dual space
is not a
Fréchet–Urysohn space and consequently, this
complete Hausdorff locally convex space
is also neither metrizable nor normable.
The strong dual space of a Fréchet space (and more generally, of bornological spaces such as metrizable TVSs) is always a complete TVS and so like any complete TVS, it is normable if and only if its topology can be induced by a complete norm (that is, if and only if it can be made into a Banach space that has the same topology). If
is a Fréchet space then
is normable if (and only if) there exists a complete
norm on its continuous dual space
such that the norm induced topology on
is
finer than the weak-* topology.
[5] Consequently, if a Fréchet space is normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.
Anderson–Kadec theorem
Note that the homeomorphism described in the Anderson–Kadec theorem is necessarily linear.
Differentiation of functions
See main article: Differentiation in Fréchet spaces. If
and
are Fréchet spaces, then the space
consisting of all continuous
linear maps from
to
is a Fréchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces, the
Gateaux derivative:
Suppose
is an open subset of a Fréchet space
is a function valued in a Fréchet space
and
The map
is
differentiable at
in the direction
if the
limitexists. The map
is said to be
continuously differentiable in
if the map
is continuous. Since the
product of Fréchet spaces is again a Fréchet space, we can then try to differentiate
and define the higher derivatives of
in this fashion.
The derivative operator
P:Cinfty([0,1])\toCinfty([0,1])
defined by
is itself infinitely differentiable. The first derivative is given by
for any two elements
This is a major advantage of the Fréchet space
over the Banach space
for finite
If
is a continuously differentiable function, then the
differential equationneed not have any solutions, and even if does, the solutions need not be unique. This is in stark contrast to the situation in Banach spaces.
In general, the inverse function theorem is not true in Fréchet spaces, although a partial substitute is the Nash–Moser theorem.
Fréchet manifolds and Lie groups
See main article: Fréchet manifold.
), and one can then extend the concept of
Lie group to these manifolds. This is useful because for a given (ordinary) compact
manifold
the set of all
diffeomorphisms
forms a generalized Lie group in this sense, and this Lie group captures the symmetries of
Some of the relations between
Lie algebras and Lie groups remain valid in this setting.
Another important example of a Fréchet Lie group is the loop group of a compact Lie group
the smooth (
) mappings
multiplied pointwise by
\left(\gamma1\gamma2\right)(t)=\gamma1(t)\gamma2(t)..
Generalizations
If we drop the requirement for the space to be locally convex, we obtain F-spaces: vector spaces with complete translation-invariant metrics.
LF-spaces are countable inductive limits of Fréchet spaces.
References
Notes and References
- Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence
in a TVS
is Cauchy if and only if for all neighborhoods
of the origin in
whenever
and
are sufficiently large. Note that this definition of a Cauchy sequence does not depend on any particular metric and doesn't even require that
be metrizable.
- Some authors do not include local convexity as part of the definition of a Fréchet space.
- Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
- Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
- Web site: The dual of a Fréchet space.. 24 February 2012. 26 April 2021.