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 not 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).[note 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 condition of locally convex 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 "F-space" and "Fréchet space" requires local convexity.
Invariant metric definitionEdit
A topological vector space is a Fréchet space if and only if it satisfies the following three properties:
- It is locally convex.[note 2]
- Its topology can 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 definitionEdit
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 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 spacesEdit
Comparison to Banach spacesEdit
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 F-norm (the F 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 spacesEdit
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
From pure functional analysisEdit
- 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 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 manifoldsEdit
- The vector space of all infinitely differentiable functions becomes a Fréchet space with the seminorms
- The vector space of all infinitely differentiable functions becomes a Fréchet space with the seminorms
- The vector space of all -times continuously differentiable functions becomes a Fréchet space with the seminorms
- 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 complete manifold 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 the space of entire (everywhere holomorphic) functions on the complex plane. Then the family of seminorms
- Let be the space of entire (everywhere holomorphic) functions of exponential type Then the family of seminorms
Properties and further notionsEdit
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 except 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 ).
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. 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 normabilityEdit
If is a locally convex space then the topology of can be a defined by a family of continuous norms on (a norm is a positive-definite seminorm) if and only if there exists at least one continuous norm on  Even if a Fréchet space has a topology that is defined by a (countable) family of norms (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 any 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.
Theorem — Let be a Fréchet space over the field Then the following are equivalent:
- does not admit a continuous norm (that is, any continuous seminorm on can not be a norm).
- contains a vector subspace that is TVS-isomorphic to
- contains a complemented vector subspace that is TVS-isomorphic to
A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space. In particular, if a locally convex metrizable space (such as a Fréchet space) is not 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. Consequently, if a Fréchet space is not normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.
Note that the homeomorphism described in the Anderson–Kadec theorem is not necessarily linear.
Eidelheit theorem — A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to
Differentiation of functionsEdit
If and are Fréchet spaces, then the space consisting of all continuous linear maps from to is not 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 limit
The derivative operator defined by is itself infinitely differentiable. The first derivative is given by
If is a continuously differentiable function, then the differential equation
Fréchet manifolds and Lie groupsEdit
One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look like Euclidean space ), 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.
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.
- Banach space – Normed vector space that is complete
- Brauner space
- Complete metric space – Metric geometry
- Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
- F-space – Topological vector space with a complete translation-invariant metric
- Fréchet lattice
- Graded Fréchet space
- Hilbert space – Generalization of Euclidean space allowing infinite dimensions
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
- Surjection of Fréchet spaces – Characterization of surjectivity
- Tame Fréchet space
- Topological vector space – Vector space with a notion of nearness
- 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.
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