Banach space

Etymology
.

Noun

 * 1)  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.
 * 2) * 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 $$X$$.
 * 1) * 2013, R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, 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 $$\left\Vert f\right\|_{X'}\equiv \text{sup}\left\{\left\vert f(x)\right\vert : \left\Vert x\right\Vert \le 1\right\}$$; $$X'$$ is also a Banach space.
 * [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 $$\left\Vert f\right\|_{X'}\equiv \text{sup}\left\{\left\vert f(x)\right\vert : \left\Vert x\right\Vert \le 1\right\}$$; $$X'$$ is also a Banach space.

Translations

 * Afrikaans: Banach-ruimte
 * Chinese:
 * Mandarin:
 * Danish: Banachrum
 * French: espace de Banach
 * German: Banach-Raum
 * Hungarian: Banach-tér
 * Italian: spazio di Banach
 * Japanese:
 * Polish: przestrzeń Banacha
 * Romanian: spațiu Banach
 * Russian:
 * Serbo-Croatian:
 * Cyrillic: Банацхов про́стор
 * Roman: Banachov próstor
 * Swedish: