Infrabarrelled space

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

Characterizations[]

If ${\displaystyle X}$ is a Hausdorff locally convex space then the canonical injection from ${\displaystyle X}$ into its bidual is a topological embedding if and only if ${\displaystyle X}$ 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[]

• Barrelled space – Topological vector space
• Quasibarrelled space
• Reflexive space
• Semi-reflexive space

References[]

Bibliography[]

• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 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. Vol. 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. Vol. 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.
• (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. Vol. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.