# 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 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.

## Properties

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

## Examples

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.

## References

- 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.