LF-space

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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 (V_n, i_{nm}) of Fréchet spaces. This means that V is a direct limit of the system (V_n, i_{nm}) in the category of locally convex topological vector spaces and each V_n is a Fréchet space.

Original definition was also assuming that V is a strict locally convex inductive limit, which means that the topology induced on V_n by V_{n+1} is identical to the original topology on V_n.

The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if U \cap V_n is an absolutely convex neighborhood of 0 in V_n for every n.

Properties[edit]

An LF space is barrelled and bornological (and thus ultrabornological).

Examples[edit]

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 \bigcup_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[edit]

  • Treves, François (1967), Topological Vector Spaces, Distributions and Kernels, Academic Press, pp. p. 126 ff .