- LF-space
In
mathematics , an "LF"-space is atopological vector space "V" that is a countable strictinductive limit ofFréchet space s. This means that for each "n" there is a subspace V_n such that:# For all "n", V_n subset V_{n+1};:# igcup_n V_n = V;:# Each V_n has a Frechet space structure;:# The topology induced on V_n by V_{n+1} is identical to the original topology on V_n.
The topology on "V" is defined by specifying that a convex subset "U" is a neighborhood of 0 if and only if U cap V_n is a neighborhood of 0 in V_n for every n.
Properties
An LF space is barrelled, bornological, and ultrabornological.
Examples
A typical example of an "LF"-space is, C^infty_c(mathbb{R}^n), the space of all infinitely differentiable functions on mathbb{R}^n with compact support. The LF-space structure is obtained by considering a sequence of compact sets K_1 subset K_2 subset ldots subset K_i subset ldots subset mathbb{R}^n with igcup_i K_i = mathbb{R}^n and for all i, K_i is a subset of the interior of K_{i+1}. Such a sequence could be the balls of radius "i" centered at the origin. The space C_c^infty(K_i) of infinitely differentiable functions on mathbb{R}^nwith compact support contained in K_i has a natural
Fréchet space structure and C^infty_c(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 K_i.With this "LF"-space structure, C^infty_c(mathbb{R}^n)is known as the space of test functions, of fundamental importance in the theory of distributions.
References
*citation|first=François|last=Treves|title=Topological Vector Spaces, Distributions and Kernels|publisher=Academic Press|year=1967|pages=p. 126 ff.
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