from
The Free On-line Dictionary of Computing (8 July 2008)
Banach space
<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)