In mathematics, an LF-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system ( V n , i n m ) of Fréchet spaces. This means that V is a direct limit of the system ( V n , i n m ) in the category of locally convex topological vector spaces and each V n 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 n by V n + 1 is identical to the original topology on V n .
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 n is an absolutely convex neighborhood of 0 in V n for every n.
An LF-space is barrelled and bornological (and thus ultrabornological).
A typical example of an LF-space is, C c ∞ ( R n ) , the space of all infinitely differentiable functions on R n with compact support. The LF-space structure is obtained by considering a sequence of compact sets K 1 ⊂ K 2 ⊂ … ⊂ K i ⊂ … ⊂ R n with ⋃ i K i = R n and for all i, K i is a subset of the interior of K i + 1 . Such a sequence could be the balls of radius i centered at the origin. The space C c ∞ ( K i ) of infinitely differentiable functions on R n with compact support contained in K i has a natural Fréchet space structure and C c ∞ ( R n ) inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets K i .
With this LF-space structure, C c ∞ ( R n ) is known as the space of test functions, of fundamental importance in the theory of distributions.
