# Banach space

## English

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### Etymology

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.