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A locally convex linear topological space displaying several properties of Banach spaces and Fréchet spaces without the metrizability condition (cf. [[Banach space|Banach space]]; [[Fréchet space|Fréchet space]]). It is one of the most extensive class of spaces to which the [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]] applies. Barrelled spaces were first introduced by N. Bourbaki.
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A [[Locally convex space|locally convex]] [[topological vector space]] displaying several properties of [[Banach space]]s and [[Fréchet space]]s without the metrizability condition. It is one of the most extensive class of spaces to which the [[Banach–Steinhaus theorem]] applies. Barrelled spaces were first introduced by N. Bourbaki.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152501.png" /> in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152502.png" /> is said to be a balanced set if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152503.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152504.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152505.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152506.png" />. A balanced set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152507.png" /> is said to be an absorbing set if it absorbs each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152508.png" />, i.e. if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b0152509.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525011.png" />.
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A set $A$ in a vector space $E$ is said to be a [[balanced set]] if $\alpha x \in A$ for all $x \in A$ and for all $\alpha$ such that $|\alpha| \le 1$. A balanced set $A$ is said to be an ''absorbing'' set if it absorbs each point of $E$, i.e. if for each $x \in E$ there exists an $\alpha > 0$ such that $\alpha x \in A$.
  
A barrel in a linear topological space is a closed, balanced, absorbing, convex set. A barrelled space is a linear topological space with a locally convex topology in which every barrel is a neighbourhood of zero. Fréchet spaces and, in particular, Banach spaces are examples of barrelled spaces. Montel spaces (cf. [[Montel space|Montel space]]) are an important class of barrelled spaces, and display remarkable properties. A quotient space of a barrelled space, a direct sum and inductive limits of barrelled spaces are barrelled spaces. Every pointwise-bounded set of continuous linear mappings of a barrelled space into a locally convex linear topological space is equicontinuous. In a space dual to a barrelled space, a bounded set in the weak topology is bounded in the strong topology and relatively compact in the weak topology.
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A barrel in a linear topological space is a closed, balanced, absorbing, convex set. A barrelled space is a linear topological space with a [[locally convex topology]] in which every barrel is a neighbourhood of zero. Fréchet spaces and, in particular, Banach spaces are examples of barrelled spaces. [[Montel space]]s are an important class of barrelled spaces, and display remarkable properties. A quotient space of a barrelled space, a direct sum and inductive limits of barrelled spaces are barrelled spaces. Every pointwise-bounded set of continuous linear mappings of a barrelled space into a locally convex topological vector space is [[Equicontinuity|equicontinuous]]. In a space dual to a barrelled space, a bounded set in the [[weak topology]] is bounded in the [[strong topology]] and relatively compact in the weak topology.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR>
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</table>
  
  
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Barrelled spaces are the most extensive class of locally convex spaces to which the Banach–Steinhaus theorem can be extended. They were first introduced in [[#References|[a4]]].
 
Barrelled spaces are the most extensive class of locally convex spaces to which the Banach–Steinhaus theorem can be extended. They were first introduced in [[#References|[a4]]].
  
A not necessarily balanced set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525013.png" /> is called absorbing if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525014.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525017.png" />. For the dual of a barrelled space the following four statements are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525018.png" /> is weakly bounded; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525019.png" /> is strongly bounded; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525020.png" /> is equi-continuous; and 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525021.png" /> is weakly compact. The last statement follows from the stronger statement that the dual of a barrelled space is quasi-complete for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015250/b01525022.png" />-topology. (For the last notion see [[Topological vector space|Topological vector space]]; [[Space of mappings, topological|Space of mappings, topological]].)
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A not necessarily balanced set $A$ in $E$ is called absorbing if for every $x \in E$ there is an $\alpha_0$ such that $x \in \alpha A$ for all $\alpha\ge \alpha_0$. For the dual of a barrelled space the following four statements are equivalent: 1) $A$ is weakly bounded; 2) $A$ is strongly bounded; 3) $A$ is equicontinuous; and 4) $A$ is weakly compact. The last statement follows from the stronger statement that the dual of a barrelled space is quasi-complete for any $\sigma$-topology. (For the last notion see [[Topological vector space]]; [[Space of mappings, topological]].)
  
 
====References====
 
====References====
<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><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Bourbaki,  "Sur certains espaces vectoriels topologiques"  ''Ann. Inst. Fourier'' , '''2'''  (1950)  pp. 5–16</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,  I. Namioka,  "Linear topological spaces" , Springer  (1963)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Bourbaki,  "Sur certains espaces vectoriels topologiques"  ''Ann. Inst. Fourier'' , '''2'''  (1950)  pp. 5–16</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 17:34, 26 September 2017

A locally convex topological vector space displaying several properties of Banach spaces and Fréchet spaces without the metrizability condition. It is one of the most extensive class of spaces to which the Banach–Steinhaus theorem applies. Barrelled spaces were first introduced by N. Bourbaki.

A set $A$ in a vector space $E$ is said to be a balanced set if $\alpha x \in A$ for all $x \in A$ and for all $\alpha$ such that $|\alpha| \le 1$. A balanced set $A$ is said to be an absorbing set if it absorbs each point of $E$, i.e. if for each $x \in E$ there exists an $\alpha > 0$ such that $\alpha x \in A$.

A barrel in a linear topological space is a closed, balanced, absorbing, convex set. A barrelled space is a linear topological space with a locally convex topology in which every barrel is a neighbourhood of zero. Fréchet spaces and, in particular, Banach spaces are examples of barrelled spaces. Montel spaces are an important class of barrelled spaces, and display remarkable properties. A quotient space of a barrelled space, a direct sum and inductive limits of barrelled spaces are barrelled spaces. Every pointwise-bounded set of continuous linear mappings of a barrelled space into a locally convex topological vector space is equicontinuous. In a space dual to a barrelled space, a bounded set in the weak topology is bounded in the strong topology and relatively compact in the weak topology.

References

[1] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)
[2] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)


Comments

Barrelled spaces are the most extensive class of locally convex spaces to which the Banach–Steinhaus theorem can be extended. They were first introduced in [a4].

A not necessarily balanced set $A$ in $E$ is called absorbing if for every $x \in E$ there is an $\alpha_0$ such that $x \in \alpha A$ for all $\alpha\ge \alpha_0$. For the dual of a barrelled space the following four statements are equivalent: 1) $A$ is weakly bounded; 2) $A$ is strongly bounded; 3) $A$ is equicontinuous; and 4) $A$ is weakly compact. The last statement follows from the stronger statement that the dual of a barrelled space is quasi-complete for any $\sigma$-topology. (For the last notion see Topological vector space; Space of mappings, topological.)

References

[a1] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
[a2] J.L. Kelley, I. Namioka, "Linear topological spaces" , Springer (1963)
[a3] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
[a4] N. Bourbaki, "Sur certains espaces vectoriels topologiques" Ann. Inst. Fourier , 2 (1950) pp. 5–16
How to Cite This Entry:
Barrelled space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Barrelled_space&oldid=14584
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article