Barrelled space

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.

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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 [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.)

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