LF-Space

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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 <math>V_n</math> such that

The topology on*V* is defined by specifying that a convex subset *U* is a neighborhood of 0 if and only if <math>U cap V_n </math> is a neighborhood of 0 in <math>V_n</math> for every n.

## Properties

An LF space is barrelled, bornological, and ultrabornological.

## Examples

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

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- # For all
*n*, <math>V_n subset V_</math>; - # <math>bigcup_n V_n = V</math>;
- # Each <math>V_n</math> has a Frechet space structure;
- # The topology induced on <math>V_n</math> by <math>V_</math> is identical to the original topology on <math>V_n</math>.

The topology on

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