Infunctional analysis, atopological vector space (TVS) is said to bequasi-complete orboundedly complete^[1] if everyclosed andbounded subset iscomplete.^[2] This concept is of considerable importance for non-metrizable TVSs.^[2]

• Every quasi-complete TVS issequentially complete.^[2]
• In a quasi-completelocally convex space, the closure of theconvex hull of a compact subset is again compact.^[3]
• In a quasi-complete Hausdorff TVS, everyprecompact subset is relatively compact.^[2]
• IfX is anormed space andY is a quasi-completelocally convex TVS then the set of allcompact linear maps ofX intoY is a closed vector subspace of${\displaystyle L_b(X;Y)}$.^[4]
• Every quasi-completeinfrabarrelled space is barreled.^[5]
• IfX is a quasi-complete locally convex space then every weakly bounded subset of the continuous dual space isstrongly bounded.^[5]
• A quasi-completenuclear space thenX has theHeine–Borel property.^[6]

Every complete TVS is quasi-complete.^[7] The product of any collection of quasi-complete spaces is again quasi-complete.^[2] The projective limit of any collection of quasi-complete spaces is again quasi-complete.^[8] Everysemi-reflexive space is quasi-complete.^[9]

The quotient of a quasi-complete space by a closed vector subspace mayfail to be quasi-complete.

There exists anLB-space that is not quasi-complete.^[10]

• Complete topological vector space – Structure in functional analysis
• Complete uniform space – Topological space with a notion of uniform propertiesPages displaying short descriptions of redirect targets

• Khaleelulla, S. M. (1982).Counterexamples in Topological Vector Spaces.Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York:Springer-Verlag.ISBN 978-3-540-11565-6.OCLC 8588370.
• 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.
• Wong, Yau-Chuen (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.

• Functional analysis

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