Infrabarrelled space

In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (sometimes spelled infrabarreled) if every bounded absorbing barrel is a neighborhood of the origin.[1]

Characterizations

If X is a Hausdorff locally convex space then the canonical injection from X into its bidual is a topological embedding if and only if X is infrabarrelled.[2]

Properties

Every quasi-complete infrabarrelled space is barrelled.[1]

Examples

Every barrelled space is infrabarrelled.[1] A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.[3]

Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.[3] Every separated quotient of an infrabarrelled space is infrabarrelled.[3]

See also

References
    1. Schaefer & Wolff 1999, p. 142.
    2. Narici & Beckenstein 2011, pp. 488–491.
    3. Schaefer & Wolff 1999, p. 194.

    Bibliography
    • Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
    • Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
    • 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.
    • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
    • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
    • Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.
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