LF-Space

# LF-space

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Description:
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 [itex]V_n[/itex] such that

# For all n, [itex]V_n subset V_[/itex];
# [itex]bigcup_n V_n = V[/itex];
# Each [itex]V_n[/itex] has a Frechet space structure;
# The topology induced on [itex]V_n[/itex] by [itex]V_[/itex] is identical to the original topology on [itex]V_n[/itex].

The topology on V is defined by specifying that a convex subset U is a neighborhood of 0 if and only if [itex]U cap V_n [/itex] is a neighborhood of 0 in [itex]V_n[/itex] for every n.

## Properties

An LF space is barrelled, bornological, and ultrabornological.

## Examples

A typical example of an LF-space is, [itex]C^infty_c(mathbb^n)[/itex], the space of all infinitely differentiable functions on [itex]mathbb^n[/itex] with compact support. The LF-space structure is obtained by considering a sequence of compact sets [itex]K_1 subset K_2 subset ldots subset K_i subset ldots subset mathbb^n[/itex] with [itex]bigcup_i K_i = mathbb^n[/itex] and for all i, [itex]K_i[/itex] is a subset of the interior of [itex]K_[/itex]. Such a sequence could be the balls of radius i centered at the origin. The space [itex]C_c^infty(K_i)[/itex] of...

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