Bornological space

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In mathematics, particularly in functional analysis, a bornological space is a locally convex space X such that every semi-norm on X which is bounded on all bounded subsets of X is continuous, where a subset A of X is bounded whenever all continuous semi-norms on X are bounded on A.

Equivalently, a locally convex space X is bornological if and only if the continuous linear operators on X to any locally convex space Y are exactly the bounded linear operators from X to Y.

For example, any metrisable locally convex space is bornological. In particular, any Fréchet space is bornological.

Given a bornological space X with continuous dual X′, then the topology of X coincides with the Mackey topology τ(X,X′). In particular, bornological spaces are Mackey spaces.