# 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 ${\displaystyle (V_{n},i_{nm})}$ of Fréchet spaces.[1] 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. The word "strict" means that each of the bonding maps ${\displaystyle i_{nm}}$ 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 ${\displaystyle V_{n}}$ by ${\displaystyle V_{n+1}}$ is identical to the original topology on ${\displaystyle V_{n}}$.[2]

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

Every LF-space is barrelled and bornological (and thus ultrabornological). Every LF-space is a meager subset of itself.[3] 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.[1] An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.[4]

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.[1]

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.[5] 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.[6] Every bounded linear operator from an LF-space into another TVS is continuous.[7]

If X is an LF-space defined by a sequence ${\displaystyle \left(X_{i}\right)_{i=1}^{\infty }}$  then the strong dual space ${\displaystyle X_{b}^{\prime }}$  of X is a Fréchet space if and only if all Xi are normable.[8] 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, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ , 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.

### Direct limit of finite-dimensional spaces

Suppose that for every positive integer n, ${\displaystyle X_{n}:=\mathbb {R} ^{n}}$  and for m < n, consider Xm as a vector subspace of Xn via the canonical embedding XmXn defined by sending ${\displaystyle x=\left(x_{1},\ldots ,x_{m}\right)\in X_{m}}$  to ${\displaystyle \left(x_{1},\ldots ,x_{m},0,\ldots ,0\right)}$ . Denote the resulting LF-space by X. The continuous dual space ${\displaystyle X^{\prime }}$  of X is equal to the algebraic dual space of X and the weak topology on ${\displaystyle X^{\prime }}$  is equal to the strong topology on ${\displaystyle X^{\prime }}$  (i.e. ${\displaystyle X_{\sigma }^{\prime }=X_{b}^{\prime }}$ ).[9] Furthermore, the canonical map of X into the continuous dual space of ${\displaystyle X_{\sigma }^{\prime }}$  is surjective.[9]

## References

1. ^ a b c Schaefer 1999, pp. 59-61.
2. ^ 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.
3. ^ Narici 2011, p. 435.
4. ^ Narici 2011, p. 436.
5. ^ Treves 2006, p. 141.
6. ^ Treves 2006, p. 173.
7. ^ Treves 2006, p. 142.
8. ^ Treves 2006, p. 201.
9. ^ a b Schaefer 1999, p. 201.