Computer Dictionary/Banach space
induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. All finite-dimensional real and complex normed vector spaces are complete and thus are Banach spaces.
Using absolute value for the norm, the real numbers are a Banach space whereas the rationals are not. This is because there are sequences of rationals that converges to irrationals.
Several theorems hold only in Banach spaces, e.g. the [[Computer Dictionary/Banach inverse mapping theorem|Banach inverse mapping theorem]]. All finite-dimensional real and complex vector spaces are Banach spaces. Hilbert spaces, spaces of integrable functions, and spaces of [[Computer Dictionary/absolutely convergent series|absolutely convergent series]] are examples of infinite-dimensional Banach spaces. Applications include wavelets, signal processing, and radar.
[Robert E. Megginson, "An Introduction to Banach Space Theory", Graduate Texts in Mathematics, 183, Springer Verlag, September 1998].