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| {{about|Fréchet spaces in functional analysis|Fréchet spaces in general topology|T1 space|the type of sequential space|Fréchet-Urysohn space}}
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| 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]] which are [[complete space|complete]] with respect to the [[metric (mathematics)|metric]] induced by the [[norm (mathematics)|norm]]). Fréchet spaces are [[locally convex space]]s which are complete with respect to a [[translation invariant metric]]. In contrast to Banach spaces, the metric need not arise from a norm.
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| Even though the [[topological structure]] of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the [[Hahn–Banach theorem]], the [[open mapping theorem (functional analysis)|open mapping theorem]], and the [[Banach–Steinhaus theorem]], still hold.
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| Spaces of [[infinitely differentiable]] [[function (mathematics)|function]]s are typical examples of Fréchet spaces.
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| == Definitions ==
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| Fréchet spaces can be defined in two equivalent ways: the first employs a [[translation-invariant metric]], the second a [[countable]] family of [[semi-norm]]s.
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| A topological vector space ''X'' is a '''Fréchet space''' if and only if it satisfies the following three properties:
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| * it is [[locally convex]]
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| * its topology can be [[induced topology|induced]] by a translation invariant metric, i.e. a metric ''d'': ''X'' × ''X'' → '''R''' such that ''d''(''x'', ''y'') = ''d''(''x''+''a'', ''y''+''a'') for all ''a'',''x'',''y'' in ''X''. This means that a subset ''U'' of ''X'' is [[open set|open]] if and only if for every ''u'' in ''U'' there exists an ε > 0 such that {''v'' : ''d''(''v'', ''u'') < ε} is a subset of ''U''.
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| * it is a [[complete space|complete]] metric space
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| Note that there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.
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| The alternative and somewhat more practical definition is the following: a topological vector space ''X'' is a '''Fréchet space''' if and only if it satisfies the following three properties:
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| * it is a [[Hausdorff space]]
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| * its topology may be induced by a countable family of semi-norms ||.||<sub>''k''</sub>, ''k'' = 0,1,2,... This means that a subset ''U'' of ''X'' is open if and only if for every ''u'' in ''U'' there exists ''K''≥0 and ε>0 such that {''v'' : ||''v'' - ''u''||<sub>''k''</sub> < ε for all ''k'' ≤ ''K''} is a subset of ''U''.
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| * it is complete with respect to the family of semi-norms
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| A sequence (''x<sub>n</sub>'') in ''X'' converges to ''x'' in the Fréchet space defined by a family of semi-norms if and only if it converges to ''x'' with respect to each of the given semi-norms.
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| ==Constructing Fréchet spaces==
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| Recall that a seminorm ǁ ⋅ ǁ is a function from a vector space ''X'' to the real numbers satisfying three properties. For all ''x'' and ''y'' in ''X'' and all scalars ''c'',
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| :<math>\|x\| \geq 0</math>
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| :<math>\|x+y\| \le \|x\| + \|y\|</math>
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| :<math>\|c\cdot x\| = |c| \|x\|</math>
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| If ǁ''x''ǁ = 0 actually implies that ''x'' = 0, then ǁ ⋅ ǁ is in fact a norm. However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows:
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| To construct a Fréchet space, one typically starts with a vector space ''X'' and defines a countable family of semi-norms ǁ ⋅ ǁ<sub>''k''</sub> on ''X'' with the following two properties:
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| * if ''x'' ∈ ''X'' and ǁ''x''ǁ<sub>''k''</sub> = 0 for all ''k'' ≥ 0, then ''x'' = 0;
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| * if (''x<sub>n</sub>'') is a sequence in ''X'' which is [[Cauchy sequence|Cauchy]] with respect to each semi-norm ǁ ⋅ ǁ<sub>''k''</sub>, then there exists ''x'' ∈ ''X'' such that (''x<sub>n</sub>'') converges to ''x'' with respect to each semi-norm ǁ ⋅ ǁ<sub>''k''</sub>.
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| Then the topology induced by these seminorms (as explained above) turns ''X'' 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 ''X'' can then be defined by
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| :<math>d(x,y)=\sum_{k=0}^\infty 2^{-k}\frac{\|x-y\|_k}{1+\|x-y\|_k} \qquad x, y \in X.</math>
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| Note that the function ''u'' → ''u''/(1+''u'') maps [0, ∞) monotonically to [0, 1), and so the above definition ensures that ''d''(''x'', ''y'') is "small" if and only if there exists ''K'' "large" such that ǁ''x'' - ''y''ǁ<sub>''k''</sub> is "small" for ''k'' = 0, …, ''K''.
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| == Examples ==
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| * 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.
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| * The [[vector space]] ''C''<sup>∞</sup>([0, 1]) of all infinitely often differentiable functions ƒ: [0,1] → '''R''' becomes a Fréchet space with the seminorms
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| ::<math>\|f\|_k = \sup\{|f^{(k)}(x)|: x \in [0,1]\}</math>
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| :for every non-negative integer ''k''. Here, ƒ<sup>''(k)''</sup> denotes the ''k''-th derivative of ƒ, and ƒ<sup>(0)</sup> = ƒ. | |
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| :In this Fréchet space, a sequence (ƒ<sub>''n''</sub>) of functions [[limit (mathematics)|converges]] towards the element ƒ of ''C''<sup>∞</sup>([0, 1]) if and only if for every non-negative integer ''k'', the sequence (<math>f_n^{(k)}</math>) [[uniform convergence|converges uniformly]] towards ƒ<sup>''(k)''</sup>.
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| * The vector space ''C''<sup>∞</sup>('''R''') of all infinitely often differentiable functions ƒ: '''R''' → '''R''' becomes a Fréchet space with the seminorms
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| ::<math> \|f\|_{k, n} = \sup \{ |f^{(k)}(x)| : x \in [-n, n] \}</math>
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| : for all integers ''k'', ''n'' ≥ 0.
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| * The vector space ''C<sup>m</sup>''('''R''') of all ''m''-times continuously differentiable functions ƒ: '''R''' → '''R''' becomes a Fréchet space with the seminorms
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| ::<math> \|f\|_{k, n} = \sup \{ |f^{(k)}(x)| : x \in [-n, n] \}</math>
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| : for all integers ''n'' ≥ 0 and ''k''=0, ...,''m''.
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| * Let ''H'' be the space of entire (everywhere [[holomorphic]]) functions on the complex plane. Then the family of seminorms
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| ::<math> \|f\|_{n} = \sup \{ |f(z)| : |z| \le n \}</math>
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| :makes ''H'' into a Fréchet space.
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| * Let ''H'' be the space of entire (everywhere holomorphic) functions of [[exponential type]] τ. Then the family of seminorms
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| ::<math> \|f\|_{n} = \sup_{z \in \mathbb{C}} \exp \left[-\left(\tau + \frac{1}{n}\right)|z|\right]|f(z)| </math>
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| :makes ''H'' into a Fréchet space.
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| * If ''M'' is a [[compact space|compact]] ''C''<sup>∞</sup>-[[manifold]] and ''B'' is a [[Banach space]], then the set ''C''<sup>∞</sup>(''M'', ''B'') of all infinitely-often differentiable functions ƒ: ''M'' → ''B'' can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If ''M'' is a (not necessarily compact) ''C''<sup>∞</sup>-manifold which admits a countable sequence ''K<sub>n</sub>'' of compact subsets, so that every compact subset of ''M'' is contained in at least one ''K<sub>n</sub>'', then the spaces ''C<sup>m</sup>''(''M'', ''B'') and ''C''<sup>∞</sup>(''M'', ''B'') are also Fréchet space in a natural manner.
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| :In fact, every smooth finite-dimensional manifold ''M'' can be made into such a nested union of compact subsets. Equip it with a [[Riemannian metric]] ''g'' which induces a metric ''d''(''x'', ''y''), choose ''x'' in ''M'', and let
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| ::<math>K_n = \{y \in M | d(x,y) \le n \} </math>
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| :Let ''M'' be a compact ''C''<sup>∞</sup>-[[manifold]] and ''V'' a [[vector bundle]] over ''M''. Let ''C''<sup>∞</sup>(''M'', ''V'') denote the space of smooth sections of ''V'' over ''X''. Choose Riemannian metrics and connections, which are guaranteed to exist, on the bundles ''TX'' and ''V''. If ''s'' is a section, denote its ''j''th covariant derivative by ''D<sup>j</sup>s''. Then
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| ::<math> \|s\|_n = \sum_{j=0}^n \sup_{x\in M}|D^js| </math>
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| :(where |⋅| is the norm induced by the Riemannian metric) is a family of seminorms making ''C''<sup>∞</sup>(''M'', ''V'') into a Fréchet space.
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| * The [[space of real valued sequences|space '''R'''<sup>ω</sup> of all real valued sequences]] becomes a Fréchet space if we define the ''k''-th semi-norm of a sequence to be the [[absolute value]] of the ''k''-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.
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| Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is the [[Lp space|space ''L<sup>p</sup>''([0, 1])]] with ''p'' < 1. This space fails to be locally convex. It is a [[F-space]].
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| == Properties and further notions ==
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| If a Fréchet space admits a continuous norm, we can take all the seminorms to be norms by adding the continuous norm to each of them. A Banach space, ''C''<sup>∞</sup>([a,b]), ''C''<sup>∞</sup>(''X'', ''V'') with ''X'' compact, and ''H'' all admit norms, while '''R'''<sup>ω</sup> and ''C''('''R''') do not.
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| 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.
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| 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]].
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| == Differentiation of functions==
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| {{main|Differentiation in Fréchet spaces}}
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| If ''X'' and ''Y'' are Fréchet spaces, then the space L(''X'',''Y'') consisting of all [[continuous function (topology)|continuous]] [[linear operator|linear maps]] from ''X'' to ''Y'' 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 [[Gâteaux derivative]]:
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| Suppose ''X'' and ''Y'' are Fréchet spaces, ''U'' is an open subset of ''X'', ''P'': ''U'' → ''Y'' is a function, ''x'' ∈ ''U'' and ''h'' ∈ ''X''. We say that ''P'' is differentiable at ''x'' in the direction ''h'' if the [[limit (mathematics)|limit]]
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| :<math>D(P)(x)(h) = \lim_{t\to 0} \,\frac{1}{t}\Big(P(x+th)-P(x)\Big)</math>
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| exists. We call ''P'' '''continuously differentiable''' in ''U'' if
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| :<math>D(P):U\times X \to Y</math>
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| is continuous. Since the [[product (topology)|product]] of Fréchet spaces is again a Fréchet space, we can then try to differentiate D(''P'') and define the higher derivatives of ''P'' in this fashion.
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| The derivative operator ''P'' : ''C''<sup>∞</sup>([0,1]) → ''C''<sup>∞</sup>([0,1]) defined by ''P''(ƒ) = ƒ′ is itself infinitely differentiable. The first derivative is given by
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| :<math>D(P)(f)(h) = h'</math>
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| for any two elements ƒ and ''h'' in ''C''<sup>∞</sup>([0,1]). This is a major advantage of the Fréchet space ''C''<sup>∞</sup>([0,1]) over the Banach space ''C<sup>k</sup>''([0,1]) for finite ''k''.
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| If ''P'' : ''U'' → ''Y'' is a continuously differentiable function, then the [[differential equation]]
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| :<math>x'(t) = P(x(t)),\quad x(0) = x_0\in U</math>
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| 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.
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| The [[inverse function theorem]] is not true in Fréchet spaces; a partial substitute is the [[Nash–Moser theorem]].
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| == Fréchet manifolds and Lie groups ==
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| {{main|Fréchet manifold}}
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| 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]] '''R'''<sup>''n''</sup>), and one can then extend the concept of [[Lie group]] to these manifolds. This is useful because for a given (ordinary) compact ''C''<sup>∞</sup> manifold ''M'', the set of all ''C''<sup>∞</sup> [[diffeomorphism]]s ƒ: ''M'' → ''M'' forms a generalized Lie group in this sense, and this Lie group captures the symmetries of ''M''. Some of the relations between [[Lie algebra]]s and Lie groups remain valid in this setting.
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| ==Generalizations==
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| If we drop the requirement for the space to be locally convex, we obtain [[F-space]]s: vector spaces with complete translation-invariant metrics.
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| [[LF-space]]s are countable inductive limits of Fréchet spaces.
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| ==References==
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| *{{springer|title=Fréchet space|id=p/f041380}}
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| *{{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Functional Analysis | publisher=McGraw-Hill Science/Engineering/Math | isbn=978-0-07-054236-5 | year=1991}}
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| *{{Citation | last1=Treves | first1=François | |authorlink1=François Treves | title=Topological vector spaces, distributions and kernels | publisher=[[Academic Press]] | location=Boston, MA | year=1967}}
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Frechet space}}
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| [[Category:Topological vector spaces]]
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| [[Category:Fréchet spaces]]
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