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In [[mathematics]], an '''''LF''-space''' is a [[topological vector space]] ''V'' that is a locally convex [[inductive limit]] of a countable inductive system <math>(V_n, i_{nm})</math> of [[Fréchet space]]s. This means that ''V'' is a [[direct limit]] of the system <math>(V_n, i_{nm})</math> in the category of [[locally convex]] topological vector spaces and each <math>V_n</math> is a Fréchet space.
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Original definition was also assuming that ''V'' is a strict locally convex inductive limit, which means that the topology induced on <math>V_n</math> by <math>V_{n+1}</math> is identical to the original topology on <math>V_n</math>.
 
The topology on ''V'' can be described by specifying that an absolutely convex subset ''U'' is a neighborhood of 0 if and only if <math>U \cap V_n </math> is an absolutely convex neighborhood of 0 in <math>V_n</math> for every n.
 
==Properties==
An LF space is [[complete]], [[barrelled space|barrelled]] and [[bornological space|bornological]] (and thus [[ultrabornological space|ultrabornological]]).
 
==Examples==
A typical example of an ''LF''-space is, <math>C^\infty_c(\mathbb{R}^n)</math>, the space of all infinitely differentiable functions on <math>\mathbb{R}^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{R}^n</math> with <math>\bigcup_i K_i = \mathbb{R}^n</math> and for all i,  <math>K_i</math> is a subset of the interior of <math>K_{i+1}</math>. Such a sequence could be the balls of radius ''i'' centered at the originThe space <math>C_c^\infty(K_i)</math> of infinitely differentiable functions on <math>\mathbb{R}^n</math>with compact support contained in <math>K_i</math> has a natural [[Fréchet space]] structure and <math>C^\infty_c(\mathbb{R}^n)</math> inherits its ''LF''-space structure as described above. The ''LF''-space topology does not depend on the particular sequence of compact sets <math>K_i</math>.
 
With this ''LF''-space structure, <math>C^\infty_c(\mathbb{R}^n)</math> is known as the space of test functions, of fundamental importance in the [[distribution (mathematics)|theory of distributions]].
 
==References==
*{{citation|first=François|last=Treves|authorlink=François Treves|title=Topological Vector Spaces, Distributions and Kernels|publisher=Academic Press|year=1967|pages=p. 126 ff}}.
 
{{Functional Analysis}}
 
[[Category:Topological vector spaces]]

Latest revision as of 03:19, 5 November 2014

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