<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].
