in mathematics, more specifically in functional analysis, a banach space (pronounced ) is a complete normed vector space. thus, a banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a cauchy sequence of vectors always converges to a well defined limit in the space. banach spaces are named after the polish mathematician stefan banach, who introduced and made a systematic study of them in 1920â1922 along with hans hahn and eduard helly. banach spaces originally grew out of the study of function spaces by hilbert, frÃ©chet, and riesz earlier in the century. banach spaces play a central role in functional analysis. in other areas of analysis, the spaces under study are often banach spaces. == definition == a banach space is a vector space  over the field  of real numbers, or over the field  of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every cauchy sequence  in , there exists an element  in  such that :\lim_{n\to\infty}x_n=x, or equivalently: :\lim_{n\to\infty}\left \|x_n - x \right \|_x = 0. the vector space structure allows one to relate the behavior of cauchy sequences to that of converging series of vectors. a normed space  is a banach space if and only if each absolutely convergent series in  converges,see theorem&nbsp;1.3.9, p.&nbsp;20 in . : \sum_{n=1}^{\infty} \|v_n\|_x < \infty \quad \text{implies that} \quad \sum_{n=1}^{\infty} v_n\ \ \text{converges in} \ \ x. completeness of a normed space is preserved if the given norm is replaced by an equivalent one. all norms on a finite-dimensional vector space are equivalent. every finite-dimensional normed space over  or  is a banach space.see corollary&nbsp;1.4.18, p.&nbsp;32 in . == general theory == === linear operators, isomorphisms ===  if  and  are normed spaces over the same ground field , the set of all continuous -linear maps  is denoted by . in infinite-dimensional spaces, not all linear maps are continuous. a linear mapping from a normed space  t
Banach_spaces Science_and_technology_in_Poland