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 $V_{n}$.

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, $C_{c}^{\infty }(\mathbb {R} ^{n})$, 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 $C_{c}^{\infty }(\mathbb {R} ^{n})$ 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, $C_{c}^{\infty }(\mathbb {R} ^{n})$ is known as the space of test functions, of fundamental importance in the theory of distributions.

References

• Treves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press, pp. 126 ff.