LF-space

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 ${\displaystyle (V_{n},i_{nm})}$ of Fréchet spaces.[1] This means that V is a direct limit of the system ${\displaystyle (V_{n},i_{nm})}$ in the category of locally convex topological vector spaces and each ${\displaystyle V_{n}}$ is a Fréchet space. The word "strict" means that each of the bonding maps ${\displaystyle i_{nm}}$ 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 ${\displaystyle V_{n}}$ by ${\displaystyle V_{n+1}}$ is identical to the original topology on ${\displaystyle V_{n}}$.[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 ${\displaystyle U\cap V_{n}}$ is an absolutely convex neighborhood of 0 in ${\displaystyle V_{n}}$ for every n.

Properties

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 X_n 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 X_n.[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 ${\displaystyle \left(X_{i}\right)_{i=1}^{\infty }}$ then the strong dual space ${\displaystyle X_{b}^{\prime }}$ of X is a Fréchet space if and only if all X_i 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.

Examples

Space of smooth compactly supported functions

Main article: Distribution (mathematics)

A typical example of an LF-space is, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$, the space of all infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support. The LF-space structure is obtained by considering a sequence of compact sets ${\displaystyle K_{1}\subset K_{2}\subset \ldots \subset K_{i}\subset \ldots \subset \mathbb {R} ^{n}}$ with ${\displaystyle \bigcup _{i}K_{i}=\mathbb {R} ^{n}}$ and for all i, ${\displaystyle K_{i}}$ is a subset of the interior of ${\displaystyle K_{i+1}}$. Such a sequence could be the balls of radius i centered at the origin. The space ${\displaystyle C_{c}^{\infty }(K_{i})}$ of infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support contained in ${\displaystyle K_{i}}$ has a natural Fréchet space structure and ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets ${\displaystyle K_{i}}$.

With this LF-space structure, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ is known as the space of test functions, of fundamental importance in the theory of distributions.

Direct limit of finite-dimensional spaces

Suppose that for every positive integer n, ${\displaystyle X_{n}:=\mathbb {R} ^{n}}$ and for m < n, consider X_m as a vector subspace of X_n via the canonical embedding X_m → X_n defined by sending ${\displaystyle x=\left(x_{1},\ldots ,x_{m}\right)\in X_{m}}$ to ${\displaystyle \left(x_{1},\ldots ,x_{m},0,\ldots ,0\right)}$. Denote the resulting LF-space by X. The continuous dual space ${\displaystyle X^{\prime }}$ of X is equal to the algebraic dual space of X and the weak topology on ${\displaystyle X^{\prime }}$ is equal to the strong topology on ${\displaystyle X^{\prime }}$ (i.e. ${\displaystyle X_{\sigma }^{\prime }=X_{b}^{\prime }}$).[9] Furthermore, the canonical map of X into the continuous dual space of ${\displaystyle X_{\sigma }^{\prime }}$ is surjective.[9]

References

1. Schaefer 1999, pp. 59-61. sfn error: no target: CITEREFSchaefer1999 (help)
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. sfn error: no target: CITEREFNarici2011 (help)
4. Narici 2011, p. 436. sfn error: no target: CITEREFNarici2011 (help)
5. Treves 2006, p. 141. sfn error: no target: CITEREFTreves2006 (help)
6. Treves 2006, p. 173. sfn error: no target: CITEREFTreves2006 (help)
7. Treves 2006, p. 142. sfn error: no target: CITEREFTreves2006 (help)
8. Treves 2006, p. 201. sfn error: no target: CITEREFTreves2006 (help)
9. Schaefer 1999, p. 201. sfn error: no target: CITEREFSchaefer1999 (help)

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)