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A complete metrizable locally convex topological vector space. Banach spaces furnish examples of Fréchet spaces, but several important function spaces are Fréchet spaces without being Banach spaces. Among these are: the Schwartz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413801.png" /> of all infinitely-differentiable complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413802.png" /> that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms
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{{MSC|46A04}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413803.png" /></td> </tr></table>
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A Fréchet space is a complete metrizable locally convex topological vector space. Banach spaces furnish examples of Fréchet spaces, but several important function spaces are Fréchet spaces without being Banach spaces. Among these are: the Schwartz space $\mathscr{S}(\R^n)$ of all infinitely-differentiable complex-valued functions on $\R^n$ that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms
 
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\[
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413805.png" /> are non-negative integer vectors; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413806.png" /> of all holomorphic functions on some open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413807.png" /> of the complex plane with the topology of uniform convergence on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041380/f0413808.png" />, etc.
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p_{\alpha,\beta}(x) = \sup_{t \in \R^n}
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\left|
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t_1^{\beta_1} \cdots t_n^{\beta_n}
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\frac{
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\partial^{\alpha_1 + \cdots + \alpha_n} x(t_1,\ldots,t_n)
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}{
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\partial t_1^{\alpha_1} \cdots \partial t_n^{\alpha_n}
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}
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\right|,
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\]
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where $\alpha$ and $\beta$ are non-negative integer vectors; the space $H(D)$ of all holomorphic functions on some open subset $D$ of the complex plane with the topology of uniform convergence on compact subsets of $D$, etc.
  
 
A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a [[Barrelled space|barrelled space]], and therefore the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach).
 
A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a [[Barrelled space|barrelled space]], and therefore the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach).
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Fréchet spaces are so named in honour of M. Fréchet.
 
Fréchet spaces are so named in honour of M. Fréchet.
  
====References====
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====References====  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"N. Bourbaki,   "Topological vector spaces" , Springer (1987) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Robertson,   W.S. Robertson,   "Topological vector spaces" , Cambridge Univ. Press  (1973)</TD></TR></table>
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{|
 
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|-
 
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|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Topological vector spaces", Springer (1987) (Translated from French)
 
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|-
====Comments====
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|valign="top"|{{Ref|KeNa}}||valign="top"| J.L. Kelley, I. Namioka, "Linear topological spaces", Springer (1963)
 
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|-
 
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|valign="top"|{{Ref|Kö}}||valign="top"| G. Köthe, "Topological vector spaces", '''1''', Springer (1969)
====References====
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|-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   I. Namioka,   "Linear topological spaces" , Springer  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>
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|valign="top"|{{Ref|RoRo}}||valign="top"| A.P. Robertson, W.S. Robertson, "Topological vector spaces", Cambridge Univ. Press (1973)
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|-
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|valign="top"|{{Ref|Sc}}||valign="top"| H.H. Schaefer, "Topological vector spaces", Macmillan (1966)
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|-
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|}

Latest revision as of 00:13, 27 July 2012

2010 Mathematics Subject Classification: Primary: 46A04 [MSN][ZBL]

A Fréchet space is a complete metrizable locally convex topological vector space. Banach spaces furnish examples of Fréchet spaces, but several important function spaces are Fréchet spaces without being Banach spaces. Among these are: the Schwartz space $\mathscr{S}(\R^n)$ of all infinitely-differentiable complex-valued functions on $\R^n$ that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms \[ p_{\alpha,\beta}(x) = \sup_{t \in \R^n} \left| t_1^{\beta_1} \cdots t_n^{\beta_n} \frac{ \partial^{\alpha_1 + \cdots + \alpha_n} x(t_1,\ldots,t_n) }{ \partial t_1^{\alpha_1} \cdots \partial t_n^{\alpha_n} } \right|, \] where $\alpha$ and $\beta$ are non-negative integer vectors; the space $H(D)$ of all holomorphic functions on some open subset $D$ of the complex plane with the topology of uniform convergence on compact subsets of $D$, etc.

A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a barrelled space, and therefore the Banach–Steinhaus theorem is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach).

Fréchet spaces are so named in honour of M. Fréchet.

References

[Bo] N. Bourbaki, "Topological vector spaces", Springer (1987) (Translated from French)
[KeNa] J.L. Kelley, I. Namioka, "Linear topological spaces", Springer (1963)
[Kö] G. Köthe, "Topological vector spaces", 1, Springer (1969)
[RoRo] A.P. Robertson, W.S. Robertson, "Topological vector spaces", Cambridge Univ. Press (1973)
[Sc] H.H. Schaefer, "Topological vector spaces", Macmillan (1966)
How to Cite This Entry:
Fréchet space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_space&oldid=27212
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article