mathematics A
complete normed
vector space. Metric is 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
Banach inverse mapping theorem. All finite-dimensional real and complex vector spaces are Banach spaces.
Hilbert spaces, spaces of integrable functions, and spaces of 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].
(2000-03-10)