Banach space

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 ${\\displaystyle H^{1}}$ and ${\\displaystyle H^{\\infty }}$, let us make a few elementary observations about the unit ball ${\\displaystyle \\Sigma }$ in the Banach space ${\\displaystyle X}$.

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