Banach space

English

Etymology

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

Noun

Banach space (plural Banach spaces)

1. (functional analysis) 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.
   • 1962 [Prentice-Hall], Kenneth Hoffman, Banach Spaces of Analytic Functions, 2007, Dover, page 138,

     Before taking up the extreme points for $H^{1}$ and $H^{\infty }$, let us make a few elementary observations about the unit ball $\Sigma$ in the Banach space $X$.

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

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

   • 2013, R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, page 35,

     [A] Banach space is a complete normed linear space $X$. Its dual space $X'$ is the linear space of all continuous linear functionals $f:X\rightarrow \mathbb {R}$, and it has norm $\left\Vert f\right\|_{X'}\equiv {\text{sup}}\left\{\left\vert f(x)\right\vert :\left\Vert x\right\Vert \leq 1\right\}$; $X'$ is also a Banach space.

Translations

complete normed vector space

• Afrikaans: Banach-ruimte
• Chinese:

  Mandarin: 巴拿赫空間／巴拿赫空间 (zh) (Bānáhè kōngjiān)

• Danish: Banachrum n
• French: espace de Banach m
• German: Banach-Raum m
• Hungarian: Banach-tér
• Italian: spazio di Banach ?
• Japanese: バナッハ空間 (ja) (Banahha-kūkan)
• Polish: przestrzeń Banacha f
• Romanian: spațiu Banach n
• Russian: ба́нахово простра́нство (ru) n (bánaxovo prostránstvo)
• Serbo-Croatian:

  Cyrillic: Банацхов про́стор m

  Roman: Banachov próstor m

• Swedish: Banachrum (sv) n

See also

• Hilbert space

Further reading

• List of Banach spaces on Wikipedia.Wikipedia