LF-space

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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

  1. For all n, V_n \subset V_{n+1};
  2. \bigcup_n V_n = V;
  3. Each Vn has a Frechet space structure;
  4. The topology induced on Vn by Vn + 1 is identical to the original topology on Vn.

The topology on V is defined by specifying that a convex subset U is a neighborhood of 0 if and only if U \cap V_n 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, C^\infty_c(\mathbb{R}^n), the space of all infinitely differentiable functions on \mathbb{R}^n with compact support. The LF-space structure is obtained by considering a sequence of compact sets K_1 \subset K_2 \subset \ldots \subset K_i \subset \ldots \subset \mathbb{R}^n with \bigcup_i K_i = \mathbb{R}^n 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 C_c^\infty(K_i) of infinitely differentiable functions on \mathbb{R}^nwith compact support contained in Ki has a natural Frechet space structure and C^\infty_c(\mathbb{R}^n) 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, C^\infty_c(\mathbb{R}^n)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.