View source for Barrelled space

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 $A$ 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 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====
<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>



====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 [[#References|[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====
<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 {{ZBL|0042.35302}}</TD></TR>
</table>

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