LF-space
From Wikipedia, the free encyclopedia
In mathematics, an LF-space is a topological vector space V that is a countable strict inductive limit of Fréchet spaces. This means that for each n there is a subspace Vn such that
-
- For all n,
; 
;- Each Vn has a Frechet space structure;
- The topology induced on Vn by Vn + 1 is identical to the original topology on Vn.
- For all n,
The topology on V is defined by specifying that a convex subset U is a neighborhood of 0 if and only if 
is a neighborhood of 0 in Vn for every n.
[edit] Properties
An LF space is barrelled, bornological, and ultrabornological.
[edit] 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, Ki is a subset of the interior of Ki + 1. 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 Ki has a natural Frechet 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 Ki.
With this LF-space structure, is known as the space of test functions, of fundamental importance in the theory of distributions.
[edit] References
- François Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967, p. 126 ff.
