Banach space


Banach space

[′bä‚näk ‚spās] (mathematics) A real or complex vector space in which each vector has a non-negative length, or norm, and in which every Cauchy sequence converges to a point of the space. Also known as complete normed linear space.

Banach Space

 

(named after S. Banach), a complete normed linear space.

Banach space

(mathematics)A complete normed vector space. Metric isinduced by the norm: d(x,y) = ||x-y||. Completeness meansthat every Cauchy sequence converges to an element of thespace. All finite-dimensional real and complex normedvector spaces are complete and thus are Banach spaces.

Using absolute value for the norm, the real numbers are aBanach space whereas the rationals are not. This is becausethere are sequences of rationals that converges toirrationals.

Several theorems hold only in Banach spaces, e.g. the Banach inverse mapping theorem. All finite-dimensional real andcomplex vector spaces are Banach spaces. Hilbert spaces,spaces of integrable functions, and spaces of absolutely convergent series are examples of infinite-dimensional Banachspaces. Applications include wavelets, signal processing,and radar.

[Robert E. Megginson, "An Introduction to Banach SpaceTheory", Graduate Texts in Mathematics, 183, Springer Verlag,September 1998].