# Barrelled space

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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; Fréchet space). 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 in a vector space is said to be a balanced set if for all and for all such that . A balanced set is said to be an absorbing set if it absorbs each point of , i.e. if for each there exists an such that .

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) 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.

#### 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)