Bornological space

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.

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