Banach space

noun

A normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.

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Meanings

nounBanach space (plural Banach spaces)

plural Banach spaces

  1. error lua execfunctional analysisA normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.

    1962 [Prentice-Hall], Kenneth Hoffman, Banach Spaces of Analytic Functions, 2007, Dover, page 138, Before taking up the extreme points for H¹ and H ᪲, let us make a few elementary observations about the unit ball Σ in the Banach space X.

    1 more example

    Already in these cases there is convergence in Banach spaces that are not only infinite-dimensional but nonseparable.

    1992, R. M. Dudley, M. G. Hahn, James Kuelbs, editors, Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference, Springer, page ix

Origin

Named after Polish mathematician Stefan Banach (1892–1945).

Full word history & related roots →

Translations

Afrikaans Banach-ruimte
  • Banach-ruimtecomplete normed vector space
Bulgarian банахово пространство
  • банахово пространство (banahovo prostranstvo)complete normed vector space
Chinese Mandarin 巴拿赫空間 /巴拿赫空间
  • 巴拿赫空間 /巴拿赫空间 (Bānáhè kōngjiān)complete normed vector space
Danish Banachrum
  • Banachrumcomplete normed vector space
Finnish Banachin avaruus
  • Banachin avaruuscomplete normed vector space
French espace de Banach
  • espace de Banachcomplete normed vector space
German Banach-Raum
  • Banach-Raumcomplete normed vector space
Hungarian Banach-tér
  • Banach-tércomplete normed vector space
Italian spazio di Banach
  • spazio di Banachcomplete normed vector space
Japanese バナッハ空間
  • バナッハ空間 (Banahha-kūkan)complete normed vector space
Polish przestrzeń Banacha
  • przestrzeń Banachacomplete normed vector space
Romanian spațiu Banach
  • spațiu Banachcomplete normed vector space
Russian ба́нахово простра́нство
  • ба́нахово простра́нство (bánaxovo prostránstvo)complete normed vector space
Swedish Banachrum
  • Banachrumcomplete normed vector space
Cite this entry
APA

Banach space. (n.d.). In WatkinsX Dictionary. Retrieved November 4, 2026, from https://dictionary.watkinsx.com/word/Banach%20space

MLA

“Banach space.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/Banach%20space. Accessed November 4, 2026.

Chicago

WatkinsX Dictionary, s.v. “Banach space,” accessed November 4, 2026, https://dictionary.watkinsx.com/word/Banach%20space.

Source: Banach space on Wiktionary, CC BY-SA 4.0.