LF-space

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 $V\_n$ such that

:# For all "n", $V\_n\; subset\; V\_\{n+1\}$;:# ;:# Each $V\_n$ has a Frechet space structure;:# The topology induced on $V\_n$ by $V\_\{n+1\}$ is identical to the original topology on $V\_n$.

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 $V\_n$ for every n.

Properties

An LF space is barrelled, bornological, and 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 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

*citation|first=François|last=Treves|title=Topological Vector Spaces, Distributions and Kernels|publisher=Academic Press|year=1967|pages=p. 126 ff.