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 $(V_n, i_{nm})$ of Fréchet spaces. This means that V is a direct limit of the system $(V_n, i_{nm})$ 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 $V_n$.^[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 $U \cap V_n$ is an absolutely convex neighborhood of 0 in $V_n$ for every n.

Properties

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

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, $K_i$ 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 $C_c^\infty(K_i)$ of infinitely differentiable functions on $\mathbb{R}^n$with compact support contained in $K_i$ has a natural Fréchet 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 $K_i$.

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.

References

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.