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 Fr├ęchet spaces. This means that for each n there is a subspace <math>V_n</math> such that

# 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 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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