Fréchet space

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,[1] 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").[1] The condition of locally convex was added later by Nicolas Bourbaki.[1] 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.[1]

DefinitionsEdit

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 definitionEdit

A topological vector space   is a Fréchet space if and only if it satisfies the following three properties:

  1. It is locally convex.[note 2]
  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  
  3. 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:

  1. It is a Hausdorff space,
  2. 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  
  3. it is complete with respect to the family of seminorms.

A family   of seminorms on   yields a Hausdorff topology if and only if[2]

 

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

Theorem[3] (de Wilde 1978) — A topological vector space   is a Fréchet space if and only if it is both a webbed space and a Baire space.

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  

ExamplesEdit

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
     
    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 sequences   converge 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 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 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 holomorphicityEdit

  • 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 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  ).[4]

Every bounded linear operator from a Fréchet space into another topological vector space (TVS) is continuous.[5]

There exists a Fréchet space   having a bounded subset   and also a dense vector subspace   such that   is not contained in the closure (in  ) of any bounded subset of  [6]

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.[7] A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual converges is strongly convergent.[7]

The strong dual space   of a Fréchet space (and more generally, of any metrizable locally convex space[8])   is a DF-space.[9] The strong dual of a DF-space is a Fréchet space.[10] The strong dual of a reflexive Fréchet space is a bornological space[8] 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.[11]

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  [12] 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.[13] The space   is not normable, which means that its topology can not be defined by any norm.[13] 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[13] — Let   be a Fréchet space over the field   Then the following are equivalent:

  1.   does not admit a continuous norm (that is, any continuous seminorm on   can not be a norm).
  2.   contains a vector subspace that is TVS-isomorphic to  
  3.   contains a complemented vector subspace that is TVS-isomorphic to  

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.[14]

A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.[9] 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.[15] Consequently, if a Fréchet space is not normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.

Anderson–Kadec theoremEdit

Anderson–Kadec theorem — Every infinite-dimensional, separable real Fréchet space is homeomorphic to   the Cartesian product of countably many copies of the real line  

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

 
exists. 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   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 equation

 
need 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 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.

Another important example of a Fréchet Lie group is the loop group of a compact Lie group   the smooth ( ) mappings   multiplied pointwise by  [16][17]

GeneralizationsEdit

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.

See alsoEdit

NotesEdit

  1. ^ 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.
  2. ^ Some authors do not include local convexity as part of the definition of a Fréchet space.

CitationsEdit

  1. ^ a b c d Narici & Beckenstein 2011, p. 93.
  2. ^ Conway 1990, Chapter 4.
  3. ^ Narici & Beckenstein 2011, p. 472.
  4. ^ Trèves 2006, pp. 166–173.
  5. ^ Trèves 2006, p. 142.
  6. ^ Wilansky 2013, p. 57.
  7. ^ a b Schaefer & Wolff 1999, pp. 194–195.
  8. ^ a b Schaefer & Wolff 1999, p. 154.
  9. ^ a b Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
  10. ^ Schaefer & Wolff 1999, p. 196.
  11. ^ Schaefer & Wolff 1999, pp. 154–155.
  12. ^ Jarchow 1981, p. 130.
  13. ^ a b c Jarchow 1981, pp. 129–130.
  14. ^ Schaefer & Wolff 1999, pp. 190–202.
  15. ^ "The dual of a Fréchet space". 24 February 2012. Retrieved 26 April 2021.
  16. ^ Sergeev 2010
  17. ^ Pressley & Segal 1986

ReferencesEdit