Open main menu

In mathematics, an LF-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that V is a direct limit of the system in the category of locally convex topological vector spaces and each 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 by is identical to the original topology on .[1]

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


An LF-space is barrelled and bornological (and thus ultrabornological).


A typical example of an LF-space is,  , the space of all infinitely differentiable functions on   with compact support. The LF-space structure is obtained by considering a sequence of compact sets   with   and for all i,   is a subset of the interior of  . Such a sequence could be the balls of radius i centered at the origin. The space   of infinitely differentiable functions on   with compact support contained in   has a natural Fréchet space structure and   inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets  .

With this LF-space structure,   is known as the space of test functions, of fundamental importance in the theory of distributions.


  1. ^ Helgason, Sigurdur (2000). Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions (Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398. ISBN 0-8218-2673-5.
  • Treves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press, pp. 126 ff.