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).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 does not even require that be metrizable.
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. 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.Some authors do not include local convexity as part of the definition of a Fréchet space. 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 exist and such that 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.
As webbed Baire spaces
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: if and for all then ; if 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
The vector space 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 seminorms
for all integers Then, a sequence of functions converges if and only if for every the sequence converges compactly.
The vector space of all -times continuously differentiable functions becomes a Fréchet space with the seminorms
for 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 let
Let 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 seminorms
makes into a Fréchet space.
Let be the space of entire (everywhere holomorphic) functions of exponential type Then the family of seminorms
makes 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 ).
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.
The strong dual space of a Fréchet space (and more generally, of any metrizable locally convex space) is a DF-space.Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) 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 cannot 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.