LB-space

In mathematics, an LB-space is a topological vector space V that is a locally convex inductive limit of a countable inductive system ${\displaystyle (V_{n},i_{nm})}$ of Banach spaces. This means that V is a direct limit of the system ${\displaystyle \left(V_{n},i_{nm}\right)}$ in the category of locally convex topological vector spaces and each ${\displaystyle V_{n}}$ is a Banach space.

Some authors restrict the term LB-space to mean that V is a strict locally convex inductive limit of Banach spaces, which means that ${\displaystyle m\leq n}$ implies ${\displaystyle V_{m}\subseteq C_{n}}$, each mapping ${\displaystyle i_{nm}:V_{m}\to V_{n}}$ is the natural inclusion, and strict means that each ${\displaystyle V_{m}}$ has ${\displaystyle V_{n}}$'s subspace topology (where ${\displaystyle m\leq n}$).[1] To distinguish this type of space from the more general definition, we will call this a strict LB-space.

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

A strict LB-space is complete,[2] barrelled,[2], and bornological[2] (and thus ultrabornological).

Examples

• If D is a locally compact topological space that is countable at infinity (i.e. equal to a countable union of compact subspaces) then the space ${\displaystyle C_{c}(D)}$ of all continuous, complex-valued functions on D with compact support is a strict LB-space.[3] For any compact subset ${\displaystyle K\subseteq D}$, let ${\displaystyle C_{c}(K)}$ denote the Banach space of complex-valued functions that are supported by K with the uniform norm and order the family of compact subsets of D by inclusion.[3]

Counter-examples

• There exists a bornological LB-space whose strong bidual is not bornological.[4]
• There exists an LB-space that is not quasi-complete.[4]

See also

• LF-space

References

1. Schaefer 1999, p. 58. sfn error: no target: CITEREFSchaefer1999 (help)
2. Schaefer 1999, pp. 60-63. sfn error: no target: CITEREFSchaefer1999 (help)
3. Schaefer 1999, pp. 57-58. sfn error: no target: CITEREFSchaefer1999 (help)
4. Khaleelulla 1982, pp. 28-63. sfn error: no target: CITEREFKhaleelulla1982 (help)

• Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• 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)