In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Fréchet spaces, named after
Maurice Fréchet, are special
topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
.
They are generalizations of
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
(
normed vector spaces that are
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
with respect to the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
induced by the
norm).
All
Banach and
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s 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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) that is
complete as a TVS, meaning that every
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
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 doesn't 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
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in .
Intuitively, a space is complete if ther ...
.
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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
" and
Banach in turn then coined the term "Fréchet space" to mean a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
, without the local convexity requirement (such a space is today often called an "
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that
# Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
").
The condition of locally convex was added later by
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook i ...
.
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
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
, the second a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
family of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s.
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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
.[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
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
.
* Assuming that the other two conditions are satisfied, this condition is equivalent to being a complete topological vector space
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
, meaning that is a complete uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
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
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
,
# 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 spaces
Comparison to Banach spaces
In contrast to
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s, 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
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
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
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
Graphs and m ...
, 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 real valued sequences, 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 manifolds
- The
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
.
- 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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
-manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
and is a Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, 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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
and a vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
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 ''j''th 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 ">, 1 with
Although this space fails to be locally convex, it is an
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that
# Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
.
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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
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
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
remain true in Fréchet spaces; examples are the
closed graph theorem
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
Graphs and m ...
and the
open mapping theorem.
The
open mapping theorem implies that if
are topologies on
that make both
and
into
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(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 space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ...
s. In the theory of stereotype spaces Fréchet spaces are dual objects to
Brauner spaces.
All
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
Montel space
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a Barrelled space, barrelled topo ...
s 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
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of a Fréchet space (and more generally, of any metrizable locally convex space)
is a
DF-space
In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
.
[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
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
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 In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
) 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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
topology on
that is
strictly coarser than this product topology.
The space
is not
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
, which means that its topology can not be defined by any
norm. Also, there does not exist
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
space is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
if and only if its
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
is a
Fréchet–Urysohn locally convex space.
[Gabriyelyan, S.S]
"On topological spaces and topological groups with certain local countable networks
(2014) 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
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
is not a
Fréchet–Urysohn space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X.
Fréchet–Urysohn spaces are a spec ...
and consequently, this
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
Hausdorff locally convex space
is also neither metrizable nor normable.
The
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
of a Fréchet space (and more generally, of
bornological spaces such as metrizable TVSs) is always a
complete TVS
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
and so like any complete TVS, it is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
that has the same topology).
If
is a Fréchet space then
is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
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 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
If
and
are Fréchet spaces, then the space
consisting of all
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
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 In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required ...
.
Fréchet manifolds and Lie groups
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
), and one can then extend the concept of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
to these manifolds.
This is useful because for a given (ordinary) compact
manifold
the set of all
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
s
forms a generalized Lie group in this sense, and this Lie group captures the symmetries of
Some of the relations between
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s 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
Generalizations
If we drop the requirement for the space to be locally convex, we obtain
F-space
In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that
# Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
s: vector spaces with complete translation-invariant metrics.
LF-space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces.
This means that ''X'' is a direct li ...
s are countable inductive limits of Fréchet spaces.
See also
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{{DEFAULTSORT:Frechet space
Topological vector spaces
F-spaces