LF-space

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In mathematics, an LF-space is a topological vector space (TVS) V that is a locally convex strict inductive limit of a countable inductive system of Fréchet spaces.[1] This means that V is a direct limit of the system in the category of locally convex topological vector spaces and each is a Fréchet space. The word "strict" means that each of the bonding maps is an embedding of TVSs.

Some authors restrict the term LF-space to mean that V is a strict locally convex inductive limit, which means that the topology induced on by is identical to the original topology on .[2]

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

PropertiesEdit

Every LF-space is barrelled and bornological (and thus ultrabornological). Every LF-space is a meager subset of itself.[3] The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete. In particular, every LF-space is complete.[1] An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.[4]

If X is the strict inductive limit of an increasing sequence of Fréchet space Xn then a subset B of X is bounded in X if and only if there exists some n such that B is a bounded subset of Xn.[1]

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.[5] A linear map from an LF-space X into a Fréchet space Y is continuous if and only if its graph is closed in X × Y.[6] Every bounded linear operator from an LF-space into another TVS is continuous.[7]

If X is an LF-space defined by a sequence   then the strong dual space   of X is a Fréchet space if and only if all Xi are normable.[8] Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.

ExamplesEdit

Space of smooth compactly supported functionsEdit

A typical example of an LF-space is,  , the space of all infinitely differentiable functions on   with compact support. The LF-space structure is obtained by considering a sequence of compact sets   with   and for all i,   is a subset of the interior of  . Such a sequence could be the balls of radius i centered at the origin. The space   of infinitely differentiable functions on   with compact support contained in   has a natural Fréchet space structure and   inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets  .

With this LF-space structure,   is known as the space of test functions, of fundamental importance in the theory of distributions.

Direct limit of finite-dimensional spacesEdit

Suppose that for every positive integer n,   and for m < n, consider Xm as a vector subspace of Xn via the canonical embedding XmXn defined by sending   to  . Denote the resulting LF-space by X. The continuous dual space   of X is equal to the algebraic dual space of X and the weak topology on   is equal to the strong topology on   (i.e.  ).[9] Furthermore, the canonical map of X into the continuous dual space of   is surjective.[9]

ReferencesEdit

  1. ^ a b c Schaefer 1999, pp. 59-61.
  2. ^ Helgason, Sigurdur (2000). Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions (Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398. ISBN 0-8218-2673-5.
  3. ^ Narici 2011, p. 435.
  4. ^ Narici 2011, p. 436.
  5. ^ Treves 2006, p. 141.
  6. ^ Treves 2006, p. 173.
  7. ^ Treves 2006, p. 142.
  8. ^ Treves 2006, p. 201.
  9. ^ a b Schaefer 1999, p. 201.