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