What is "lf-space"

In mathematics, an LF-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system (V, i) of Fréchet spaces. This means that V is a direct limit of the system (V, i) in the category of locally convex topological vector spaces and each V is a Fréchet space.

Some authors restrict the term LF-space to mean that V is a strict locally convex inductive limit, which means that the topology induced on V by V is identical to the original topology on V.

The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if U ∩ V is an absolutely convex neighborhood of 0 in V for every n.