# 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 ${\displaystyle (V_{n},i_{nm})}$ of Fréchet spaces. This means that V is a direct limit of the system ${\displaystyle (V_{n},i_{nm})}$ in the category of locally convex topological vector spaces and each ${\displaystyle V_{n}}$ is a Fréchet space.

Original definition was also assuming that V is a strict locally convex inductive limit, which means that the topology induced on ${\displaystyle V_{n}}$ by ${\displaystyle V_{n+1}}$ is identical to the original topology on ${\displaystyle V_{n}}$.

The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if ${\displaystyle U\cap V_{n}}$ is an absolutely convex neighborhood of 0 in ${\displaystyle V_{n}}$ for every n.

## Properties

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

## Examples

A typical example of an LF-space is, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$, the space of all infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support. The LF-space structure is obtained by considering a sequence of compact sets ${\displaystyle K_{1}\subset K_{2}\subset \ldots \subset K_{i}\subset \ldots \subset \mathbb {R} ^{n}}$ with ${\displaystyle \bigcup _{i}K_{i}=\mathbb {R} ^{n}}$ and for all i, ${\displaystyle K_{i}}$ is a subset of the interior of ${\displaystyle K_{i+1}}$. Such a sequence could be the balls of radius i centered at the origin. The space ${\displaystyle C_{c}^{\infty }(K_{i})}$ of infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$with compact support contained in ${\displaystyle K_{i}}$ has a natural Fréchet space structure and ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets ${\displaystyle K_{i}}$.

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

## References

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