# LF-space

(Redirected from LF space)

In mathematics, an LF-space is a topological vector space (TVS) V that is a locally convex strict 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. The word "strict" means that each of the bonding maps $i_{nm}$ is an embedding of TVSs.

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

Every LF-space is barrelled and bornological (and thus ultrabornological). Every LF-space is a meager subset of itself. The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete. In particular, every LF-space is complete. An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.

If X is the strict inductive limit of an increasing sequence of Fréchet space Xn then a subset B of X is bounded in X if and only if there exists some n such that B is a bounded subset of Xn.

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous. A linear map from an LF-space X into a Fréchet space Y is continuous if and only if its graph is closed in X × Y. Every bounded linear operator from an LF-space into another TVS is continuous.

If X is an LF-space defined by a sequence $\left(X_{i}\right)_{i=1}^{\infty }$  then the strong dual space $X_{b}^{\prime }$  of X is a Fréchet space if and only if all Xi are normable. Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.

## Examples

### Space of smooth compactly supported functions

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.

### Direct limit of finite-dimensional spaces

Suppose that for every positive integer n, $X_{n}:=\mathbb {R} ^{n}$  and for m < n, consider Xm as a vector subspace of Xn via the canonical embedding XmXn defined by sending $x=\left(x_{1},\ldots ,x_{m}\right)\in X_{m}$  to $\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right)$ . Denote the resulting LF-space by X. The continuous dual space $X^{\prime }$  of X is equal to the algebraic dual space of X and the weak topology on $X^{\prime }$  is equal to the strong topology on $X^{\prime }$  (i.e. $X_{\sigma }^{\prime }=X_{b}^{\prime }$ ). Furthermore, the canonical map of X into the continuous dual space of $X_{\sigma }^{\prime }$  is surjective.