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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 Banachinverse mapping theorem. All finite-dimensional real andcomplex vector spaces are Banach spaces. Hilbert spaces,spaces of integrable functions, and spaces of absolutelyconvergent 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].