View source for Fréchet space

{{Short description|A locally convex topological vector space that is also a complete metric space}}
{{about|Fréchet spaces in functional analysis|Fréchet spaces in general topology|T1 space|the type of sequential space|Fréchet–Urysohn 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 space|complete]] with respect to the [[Metric (mathematics)|metric]] induced by the [[Norm (mathematics)|norm]]). 
All [[Banach space|Banach]] and [[Hilbert space]]s are Fréchet spaces. 
Spaces of [[infinitely differentiable]] [[Function (mathematics)|function]]s are typical examples of Fréchet spaces, many of which are typically {{em|not}} Banach spaces. 

A Fréchet space <math>X</math> is defined to be a [[Locally convex topological vector space|locally convex]] [[Metrizable topological vector space|metrizable]] [[topological vector space]] (TVS) that is [[Complete topological vector space|complete as a TVS]],{{sfn|Narici|Beckenstein|2011|p=93}} meaning that every [[Cauchy sequence]] in <math>X</math> converges to some point in <math>X</math> (see footnote for more details).<ref group=note>Here "Cauchy" means Cauchy with respect to the [[Complete topological vector space|canonical uniformity]] that every [[Topological vector space|TVS]] possess. That is, a sequence <math>x_{\bull} = \left(x_m\right)_{m=1}^{\infty}</math> in a TVS <math>X</math> is Cauchy if and only if for all neighborhoods <math>U</math> of the origin in <math>X,</math> <math>x_m - x_n \in U</math> whenever <math>m</math> and <math>n</math> are sufficiently large. Note that this definition of a Cauchy sequence does not depend on any particular metric and doesn't even require that <math>X</math> be metrizable.</ref> 

:'''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 space|complete metric]]. 
Conversely, if the topology of a locally convex space <math>X</math> is induced by a translation-invariant complete metric then <math>X</math> is a Fréchet space. 

[[Maurice René Fréchet|Fréchet]] was the first to use the term "[[Banach space]]" and [[Stefan Banach|Banach]] in turn then coined the term "Fréchet space" to mean a [[Complete topological vector space|complete]] [[metrizable topological vector space]], without the local convexity requirement (such a space is today often called an "[[F-space]]").{{sfn|Narici|Beckenstein|2011|p=93}} 
The local convexity requirement was added later by [[Nicolas Bourbaki]].{{sfn|Narici|Beckenstein|2011|p=93}} 
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 "{{em|F}}-space" and "Fréchet space" requires local convexity.{{sfn|Narici|Beckenstein|2011|p=93}} 

==Definitions==

Fréchet spaces can be defined in two equivalent ways: the first employs a [[translation-invariant]] [[Metric (mathematics)|metric]], the second a [[countable]] family of [[seminorm]]s.

===Invariant metric definition===

A topological vector space <math>X</math> is a '''Fréchet space''' if and only if it satisfies the following three properties:
<ol>
<li>It is [[locally convex]].<ref group=note>Some authors do not include local convexity as part of the definition of a Fréchet space.</ref></li>
<li>Its topology '''{{em|can}}''' be [[Induced topology|induced]] by a translation-invariant metric, that is, a metric <math>d : X \times X \to \R</math> such that <math>d(x, y) = d(x + z, y + z)</math> for all <math>x, y, z \in X.</math> This means that a subset <math>U</math> of <math>X</math> is [[Open set|open]] if and only if for every <math>u \in U</math> there exists an <math>r > 0</math> such that <math>\{v : d(v, u) < r\}</math> is a subset of <math>U.</math></li>
<li>Some (or equivalently, every) translation-invariant metric on <math>X</math> inducing the topology of  <math>X</math> is [[complete metric|complete]]. 
* Assuming that the other two conditions are satisfied, this condition is equivalent to <math>X</math> being a [[complete topological vector space]], meaning that <math>X</math> is a complete [[uniform space]] when it is endowed with its [[Complete topological vector space|canonical uniformity]] (this canonical uniformity is independent of any metric on <math>X</math> and is defined entirely in terms of vector subtraction and <math>X</math>'s neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on <math>X</math> is identical to this canonical uniformity).</li>
</ol>

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 <math>X</math> 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 <math>(\|\cdot\|_k)_{k \in \N_0}</math>. This means that a subset <math>U \subseteq X</math> is open if and only if for every <math>u \in U</math> there exist <math>K \geq 0</math> and <math>r > 0</math> such that <math>\{ v \in X : \|v-u\|_k < r \text{ for all } k \le K\}</math> is a subset of <math>U</math>.
# It is complete with respect to the family of seminorms. 

A family <math>\mathcal P</math> of seminorms on <math>X</math> yields a Hausdorff topology if and only if<ref>{{harvnb|Conway|1990|loc=Chapter 4.}}</ref>
<math display=block>\bigcap_{\|\cdot\| \in \mathcal{P}} \{x \in X : \|x\| = 0\} = \{0\}.</math>

A sequence <math>\left(x_n\right)_{n \in \N}</math> in <math>X</math> converges to <math>x</math> in the Fréchet space defined by a family of seminorms if and only if it converges to <math>x</math> with respect to each of the given seminorms.

===As webbed Baire spaces===

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|p=472}}|note=de Wilde 1978|math_statement=
A [[topological vector space]] <math>X</math> is a Fréchet space if and only if it is both a [[webbed space]] and a [[Baire space]].
}}

===Comparison to Banach spaces===

In contrast to [[Banach space]]s, the complete translation-invariant metric need not arise from a norm. 
The topology of a Fréchet space does, however, arise from both a [[Paranorm|total paranorm]] and an [[F-seminorm|{{em|F}}-norm]] (the {{em|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 (functional analysis)|open mapping theorem]], the [[closed graph theorem]], and the [[Banach–Steinhaus theorem]], still hold.

==Constructing Fréchet spaces==

Recall that a seminorm <math>\|\cdot\|</math> is a function from a vector space <math>X</math> to the real numbers satisfying three properties. 
For all <math>x, y \in X</math> and all scalars <math>c,</math>
<math display=block>\|x\| \geq 0</math>
<math display=block>\|x+y\| \le \|x\| + \|y\|</math>
<math display=block>\|c\cdot x\| = |c|\|x\|</math>

If <math>\| x\| = 0 \iff x = 0</math>, then <math>\|\cdot\|</math> 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 <math>X</math> and defines a countable family of seminorms  <math>\|\cdot\|_k</math> on <math>X</math> with the following two properties:
* if <math>x \in X</math> and <math>\|x\|_k = 0</math> for all <math>k \geq 0,</math> then <math>x = 0</math>;
* if <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> is a sequence in <math>X</math> which is [[Cauchy sequence|Cauchy]] with respect to each seminorm <math>\|\cdot\|_k,</math> then there exists <math>x \in X</math> such that <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> converges to <math>x</math> with respect to each seminorm <math>\|\cdot\|_k.</math>

Then the topology induced by these seminorms (as explained above) turns <math>X</math> 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 <math>X</math> can then be defined by
<math display=block>d(x,y)=\sum_{k=0}^\infty 2^{-k}\frac{\|x-y\|_k}{1+\|x-y\|_k} \qquad x, y \in X.</math>

The function <math>u \mapsto \frac{u}{1 + u}</math> maps <math>[0, \infty)</math> monotonically to <math>[0, 1),</math> and so the above definition ensures that <math>d(x, y)</math> is "small" if and only if there exists <math>K</math> "large" such that <math>\|x - y\|_k</math> is "small" for <math>k = 0, \ldots, K.</math>

==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 <math>\R^{\omega}</math> of all real valued sequences]] (also denoted <math>\R^{\N}</math>) becomes a Fréchet space if we define the <math>k</math>-th seminorm of a sequence to be the [[absolute value]] of the <math>k</math>-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.

===From smooth manifolds===

<ul>
<li>The [[vector space]] <math>C^\infty([0,1])</math> of all infinitely differentiable functions <math>f : [0,1] \to \R</math> becomes a Fréchet space with the seminorms
<math display=block>\|f\|_k = \sup\{|f^{(k)}(x)| : x \in [0,1]\}</math>
for every non-negative integer <math>k.</math> Here, <math>f^{(k)}</math> denotes the <math>k</math>-th derivative of <math>f,</math> and <math>f^{(0)} = f.</math> In this Fréchet space, a sequence <math>\left(f_n\right) \to f</math> of functions [[Limit (mathematics)|converges]] towards the element <math>f \in C^\infty([0,1])</math> if and only if for every non-negative integer <math>k \geq 0,</math> the sequence <math>\left(f_n^{(k)}\right) \to f^{(k)}</math> [[Uniform convergence|converges uniformly]].</li>

<li>The vector space <math>C^\infty(\R)</math> of all infinitely differentiable functions <math>f : \R \to \R</math> becomes a Fréchet space with the seminorms
<math display=block>\|f\|_{k, n} = \sup \{ |f^{(k)}(x)| : x \in [-n, n]\}</math>
for all integers <math>k, n \geq 0.</math> Then, a sequence of functions <math>\left(f_n\right) \to f</math> converges if and only if for every <math>k, n \geq 0,</math> the sequences <math>\left(f_n^{(k)}\right) \to f^{(k)}</math> [[Compact convergence|converge compactly]].</li>

<li>The vector space <math>C^m(\R)</math> of all <math>m</math>-times continuously differentiable functions <math>f : \R \to \R</math> becomes a Fréchet space with the seminorms
<math display=block>\|f\|_{k, n} = \sup \{ |f^{(k)}(x)| : x \in [-n, n]\}</math>
for all integers <math>n \geq 0</math> and <math>k = 0, \ldots, m.</math></li>

<li>If <math>M</math> is a [[Compact space|compact]] <math>C^{\infty}</math>-[[manifold]] and <math>B</math> is a [[Banach space]], then the set <math>C^{\infty}(M, B)</math> of all infinitely-often differentiable functions <math>f : M \to B</math> can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If <math>M</math> is a (not necessarily compact) <math>C^{\infty}</math>-manifold which admits a countable sequence <math>K^n</math> of compact subsets, so that every compact subset of <math>M</math> is contained in at least one <math>K^n,</math> then the spaces <math>C^m(M, B)</math> and <math>C^{\infty}(M, B)</math> are also Fréchet space in a natural manner.

As a special case, every smooth finite-dimensional {{em|[[Geodesic manifold|complete manifold]]}} <math>M</math> can be made into such a nested union of compact subsets: equip it with a [[Riemannian metric]] <math>g</math> which induces a metric <math>d(x, y),</math> choose <math>x \in M,</math> and let
<math display=block>K_n = \{y \in M : d(x,y) \le n\} \ .</math>

Let <math>X</math> be a compact <math>C^{\infty}</math>-[[manifold]] and<math>V</math> a [[vector bundle]] over <math>X.</math>  Let <math>C^{\infty}(X, V)</math> denote the space of smooth sections of <math>V</math> over <math>X.</math>  Choose Riemannian metrics <math>g</math> and connections <math>D</math>, which are guaranteed to exist, on the bundles <math>TX</math> and <math>V.</math>  If <math>s</math> is a section, denote its ''j''<sup>th</sup> covariant derivative by <math>D^j s.</math>  Then
<math display=block>\|s\|_n = \sum_{j=0}^n \sup_{x\in M} |D^js|_g</math>
(where <math>|\,\cdot\,|_g</math> is the norm induced by the Riemannian metric <math>g</math>) is a family of seminorms making <math>C^{\infty}(M, V)</math> into a Fréchet space.</li>
</ul>

===From holomorphicity===

<ul>
<li>Let <math>H</math> be the space of entire (everywhere [[Holomorphic function|holomorphic]]) functions on the complex plane.  Then the family of seminorms
<math display=block>\left|f\right|_{n} = \sup \{\left|f(z)\right| : |z| \leq n\}</math>
makes <math>H</math> into a Fréchet space.</li>

<li>Let <math>H</math> be the space of entire (everywhere holomorphic) functions of [[exponential type]] <math>\tau.</math>  Then the family of seminorms
<math display=block>\left|f\right|_{n} = \sup_{z \in \Complex} \exp \left[- \left(\tau + \frac{1}{n} \right) |z|\right] \left|f(z)\right|</math>
makes <math>H</math> into a Fréchet space.</li>
</ul>

Not all vector spaces with complete translation-invariant metrics are Fréchet spaces.  An example is the [[Lp space|space <math>L^p([0, 1])</math>]] with <math>p < 1.</math> 
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, <math>C^{\infty}([a, b]),</math> <math>C^{\infty}(X, V)</math> with <math>X</math> compact, and <math>H</math> all admit norms, while <math>\R^{\omega}</math> and <math>C(\R)</math> 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 [[Countable set|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 {{em|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 <math>I \neq \varnothing</math> is any set and <math>X</math> is any non-trivial Fréchet space (such as <math>X = \R</math> for instance), then the product <math>X^I = \prod_{i \in I} X</math> is a Fréchet space if and only if <math>I</math> 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 (functional analysis)|open mapping theorem]]. 
The [[Open mapping theorem (functional analysis)|open mapping theorem]] implies that if <math>\tau \text{ and } \tau_2</math> are topologies on <math>X</math> that make both <math>(X, \tau)</math> and <math>\left(X, \tau_2\right)</math> into [[Complete topological vector space|complete]] [[Metrizable topological vector space|metrizable TVSs]] (such as Fréchet spaces) and if one topology is [[Comparison of topologies|finer or coarser]] than the other then they must be equal (that is, if <math>\tau \subseteq \tau_2 \text{ or } \tau_2 \subseteq \tau \text{ then } \tau = \tau_2</math>).{{sfn|Trèves|2006|pp=166–173}}

Every [[Bounded linear map|bounded]] linear operator from a Fréchet space into another [[topological vector space]] (TVS) is continuous.{{sfn|Trèves|2006|p=142}} 

There exists a Fréchet space <math>X</math> having a [[Bounded set (topological vector space)|bounded]] subset <math>B</math> and also a dense vector subspace <math>M</math> such that <math>B</math> is {{em|not}} contained in the closure (in <math>X</math>) of any bounded subset of <math>M.</math>{{sfn|Wilansky|2013|p=57}} 

All Fréchet spaces are [[stereotype space]]s. In the theory of stereotype spaces Fréchet spaces are dual objects to [[Brauner space]]s. 
All [[Metrizable topological vector space|metrizable]] [[Montel space]]s are [[Separable space|separable]].{{sfn|Schaefer|Wolff|1999|pp=194-195}} A [[Separable space|separable]] Fréchet space is a Montel space if and only if each [[Weak-* topology|weak-* convergent]] sequence in its continuous dual converges is [[Strong dual space|strongly convergent]].{{sfn|Schaefer|Wolff|1999|pp=194-195}} 

The [[strong dual space]] <math>X_b^{\prime}</math> of a Fréchet space (and more generally, of any metrizable locally convex space{{sfn|Schaefer|Wolff|1999|p=154}}) <math>X</math> is a [[DF-space]].<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref> 
The strong dual of a DF-space is a Fréchet space.{{sfn|Schaefer|Wolff|1999|p=196}} 
The strong dual of a [[Reflexive space|reflexive]] Fréchet space is a [[bornological space]]{{sfn|Schaefer|Wolff|1999|p=154}} 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.{{sfn|Schaefer|Wolff|1999|pp=154-155}} 

===Norms and normability===

{{See also|Metrizable topological vector space#Normability}}

If <math>X</math> is a locally convex space then the topology of <math>X</math> can be a defined by a family of continuous {{em|norms}} on <math>X</math> (a '''[[Norm (mathematics)|norm]]''' is a [[Positive-definite functional|positive-definite]] [[seminorm]]) if and only if there exists {{em|at least one}} continuous {{em|norm}} on <math>X.</math>{{sfn|Jarchow|1981|p=130}} 
Even if a Fréchet space has a topology that is defined by a (countable) family of {{em|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 real sequences|space of all sequences]] <math>\mathbb{K}^{\N}</math> (with the product topology) is a Fréchet space. There does not exist any Hausdorff [[Locally convex topological vector space|locally convex]] topology on <math>\mathbb{K}^{\N}</math> that is [[Comparison of topologies|strictly coarser]] than this product topology.{{sfn|Jarchow|1981|pp=129-130}} 
The space <math>\mathbb{K}^{\N}</math> is not [[Normable space|normable]], which means that its topology can not be defined by any [[Norm (mathematics)|norm]].{{sfn|Jarchow|1981|pp=129-130}} Also, there does not exist {{em|any}} [[Continuous function|continuous]] norm on <math>\mathbb{K}^{\N}.</math> In fact, as the following theorem shows, whenever <math>X</math> is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to the presence of <math>\mathbb{K}^{\N}</math> as a subspace. 

{{Math theorem|name=Theorem{{sfn|Jarchow|1981|pp=129-130}}|math_statement=
Let <math>X</math> be a Fréchet space over the field <math>\mathbb{K}.</math> 
Then the following are equivalent:
<ol>
<li><math>X</math> does {{em|not}} admit a continuous norm (that is, any continuous seminorm on <math>X</math> can {{em|not}} be a norm).</li>
<li><math>X</math> contains a vector subspace that is TVS-isomorphic to <math>\mathbb{K}^{\N}.</math></li>
<li><math>X</math> contains a [[Complemented subspace|complemented vector subspace]] that is TVS-isomorphic to <math>\mathbb{K}^{\N}.</math></li>
</ol>
}}

If <math>X</math> is a non-normable Fréchet space on which there exists a continuous norm, then <math>X</math> contains a closed vector subspace that has no [[topological complement]].{{sfn|Schaefer|Wolff|1999|pp=190-202}} 

A metrizable [[locally convex]] space is [[Normable space|normable]] if and only if its [[strong dual space]] is a [[Fréchet–Urysohn space|Fréchet–Urysohn]] locally convex space.<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref> In particular, if a locally convex metrizable space <math>X</math> (such as a Fréchet space) is {{em|not}} normable (which can only happen if <math>X</math> is infinite dimensional) then its [[strong dual space]] <math>X^{\prime}_b</math> is not a [[Fréchet–Urysohn space]] and consequently, this [[Complete topological vector space|complete]] Hausdorff locally convex space <math>X^{\prime}_b</math> is also neither metrizable nor normable.

The [[strong dual space]] of a Fréchet space (and more generally, of [[bornological space]]s such as metrizable TVSs) is always a [[Complete topological vector space|complete TVS]] and so like any complete TVS, it is [[Normable space|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 <math>X</math> is a Fréchet space then <math>X</math> is [[Normable space|normable]] if (and only if) there exists a complete [[Norm (mathematics)|norm]] on its continuous dual space <math>X'</math> such that the norm induced topology on <math>X'</math> is [[Comparison of topologies|finer]] than the weak-* topology.<ref>{{cite web|title=The dual of a Fréchet space.|url=https://math.stackexchange.com/a/112852/239329|date=24 February 2012|access-date=26 April 2021}}</ref> 
Consequently, if a Fréchet space is {{em|not}} normable (which can only happen if it is infinite dimensional) then neither is its strong dual space. 

===Anderson–Kadec theorem===

{{Math theorem|name=[[Anderson–Kadec theorem]]|math_statement=
Every infinite-dimensional, separable real Fréchet space is homeomorphic to <math>\R^{\N},</math> the [[Cartesian product]] of [[Countable set|countably many]] copies of the real line <math>\R.</math>
}}

Note that the homeomorphism described in the Anderson–Kadec theorem is {{em|not}} necessarily linear. 

{{Math theorem|name=[[Meier Eidelheit|Eidelheit]] theorem|math_statement=
A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to <math>\R^{\N}.</math>
}}

==Differentiation of functions==
{{main|Differentiation in Fréchet spaces}}
 
If <math>X</math> and <math>Y</math> are Fréchet spaces, then the space <math>L(X,Y)</math> consisting of all [[Continuous function (topology)|continuous]] [[Linear operator|linear maps]] from <math>X</math> to <math>Y</math> is {{em|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 <math>U</math> is an open subset of a Fréchet space <math>X,</math> <math>P : U \to Y</math> is a function valued in a Fréchet space <math>Y,</math> <math>x \in U</math> and <math>h \in X.</math> The map <math>P</math> is '''differentiable at <math>x</math> in the direction <math>h</math>''' if the [[Limit (mathematics)|limit]]
<math display=block>D(P)(x)(h) = \lim_{t \to 0} \,\frac{1}{t}\left(P(x + th) - P(x)\right)</math>
exists. 
The map <math>P</math> is said to be '''continuously differentiable''' in <math>U</math> if the map
<math display=block>D(P) : U \times X \to Y</math>
is continuous. Since the [[Product (topology)|product]] of Fréchet spaces is again a Fréchet space, we can then try to differentiate <math>D(P)</math> and define the higher derivatives of <math>P</math> in this fashion. 

The derivative operator <math>P : C^{\infty}([0, 1]) \to C^{\infty}([0, 1])</math> defined by <math>P(f) = f'</math> is itself infinitely differentiable. The first derivative is given by
<math display=block>D(P)(f)(h) = h'</math>
for any two elements <math>f, h \in C^{\infty}([0, 1]).</math> 
This is a major advantage of the Fréchet space <math>C^{\infty}([0, 1])</math> over the Banach space <math>C^k([0, 1])</math> for finite <math>k.</math> 

If <math>P : U \to Y</math> is a continuously differentiable function, then the [[differential equation]]
<math display=block>x'(t) = P(x(t)),\quad x(0) = x_0\in U</math>
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 groups==
{{main|Fréchet manifold}}

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]] <math>\R^n</math>), and one can then extend the concept of [[Lie group]] to these manifolds. 
This is useful because for a given (ordinary) compact <math>C^{\infty}</math> manifold <math>M,</math> the set of all <math>C^{\infty}</math> [[diffeomorphism]]s <math>f : M \to M</math> forms a generalized Lie group in this sense, and this Lie group captures the symmetries of <math>M.</math> 
Some of the relations between [[Lie algebra]]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 <math>G,</math> the smooth (<math>C^{\infty}</math>) mappings <math>\gamma : S^1 \to G,</math> multiplied pointwise by <math>\left(\gamma_1 \gamma_2\right)(t) = \gamma_1(t) \gamma_2(t)..</math><ref>{{harvnb|Sergeev|2010}}</ref><ref>{{harvnb|Pressley|Segal|1986}}</ref>

==Generalizations==

If we drop the requirement for the space to be locally convex, we obtain [[F-space]]s: vector spaces with complete translation-invariant metrics.

[[LF-space]]s are countable inductive limits of Fréchet spaces.

==See also==

* {{annotated link|Banach space}}
* {{annotated link|Brauner space}}
* {{annotated link|Complete metric space}}
* {{annotated link|Complete topological vector space}}
* {{annotated link|F-space}}
* {{annotated link|Fréchet lattice}}
* {{annotated link|Graded Fréchet space}}
* {{annotated link|Hilbert space}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Surjection of Fréchet spaces}}
* {{annotated link|Tame Fréchet space}}
* {{annotated link|Topological vector space}}

==Notes==

{{reflist|group=note}}

==Citations==

{{reflist}}

==References==

* {{springer|title=Fréchet space|id=p/f041380}}
* {{Berberian Lectures in Functional Analysis and Operator Theory}} <!-- {{sfn|Berberian|2014|p=}} -->
* {{Bourbaki Topological Vector Spaces}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{Conway A Course in Functional Analysis}} <!-- {{sfn|Conway|1990|p=}} -->
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn|Edwards|1995|p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn|Grothendieck|1973|p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1983|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{cite book|last1=Pressley|first1=Andrew|last2=Segal|first2=Graeme|authorlink2=Graeme Segal|title=Loop groups|series=Oxford Mathematical Monographs. Oxford Science Publications|publisher=[[Oxford University Press]]|location=New York|year=1986|isbn=0-19-853535-X|mr=0900587|url=https://books.google.com/books?id=MbFBXyuxLKgC}}
* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|1964|p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{cite book|last=Sergeev|first=Armen|title=Kähler Geometry of Loop Spaces|year=2010|series=Mathematical Society of Japan Memoirs|volume=23|isbn=978-4-931469-60-0|publisher=[[World Scientific Publishing]]|doi=10.1142/e023|url=http://www.worldscientific.com/worldscibooks/10.1142/e023}}
* {{Adasch Topological Vector Spaces}} <!-- {{sfn|Adasch|1978|p=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional Analysis}}
{{TopologicalVectorSpaces}}

{{DEFAULTSORT:Frechet space}}
[[Category:Fréchet spaces| ]]
[[Category:Topological vector spaces]]
[[Category:F-spaces]]